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Thermodynamics and rheology of partially miscible polymer blends Chopra, Divya 2002

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THERMODYNAMICS AND RHEOLOGY OF PARTIALLY MISCD3LE POLYMER BLENDS by DIVYA CHOPRA M.A.Sc, Chemical Engineering, University of British Columbia, 1998 B.Tech., Chemical Technology (Plastics), Harcourt Butler Technological Institute, India, 1995 A THESIS SUBMITTED IN P A R T I A L F U L F I L L M E N T OF THE REQUIREMENTS FOR THE D E G R E E OF DOCTOR OF PHILOSOPHY in the Faculty of Graduate Studies Department of Chemical and Bio-Resource Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 2002 ©Divya Chopra, 2002 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ^ H f M l c f r v - ^ £ j p LOO>IUH, tv£E£JMC, _ The University of British Columbia Vancouver, Canada Date DE-6 (2/88) T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S ii Abstract The phase behaviour of two partially miscible binary polymer blend systems was investigated using rheological, optical, turbidimetric and light-scattering techniques. Rheological techniques included the evaluation of linear and non-linear rheology using parallel plate, sliding plate and capillary rheometry. The two systems comprised a) a model upper critical solution temperature mixture of varying M w (4.3-162 kg/mol) poly(dimethylsiloxane) and high molecular weight (Mw = 83.8 kg/mol) poly(ethylmethylsiloxane) [PDMS/PEMS]; and b) a lower critical solution temperature mixture of high M w (100-200 kg/mol) poly(styrene-co-maleic anhydride) and poly(methyl methacrylate) [SMA/PMMA] with SMA component having 8%, 14% and 32% by weight maleic anhydride content (MA). The Theologically determined phase separation temperature was marked by changes in the temperature dependence of the elastic moduli at constant frequency and of the stresses at constant shear rate. Failure of time-temperature superposition in frequency sweeps in the linear region was also observed. The phase diagrams of the blends were modeled by using a temperature dependent expression for the interaction parameter of Flory-Huggins theory, based on the concept of generalized Gibbs free energy of mixing. The phase behaviour, morphology and interfacial tension of PDMS/PEMS blends was very sensitive to the molecular weight of the individual components. Miscibility increased and interfacial tension decreased as the molecular weight of PDMS decreased. SMA/PMMA blends containing 14% M A were found to be more miscible than those containing 8% or 32% MA, a finding attributed to the compositional dependence of intermolecular (SMA-SMA) and intramolecular (SMA-PMMA) interactions in the different samples. Shear-induced fibrillar type growth of SMA inclusions in the phase-separated region in the case of SMA/PMMA with 32%MA was established using rheological tools. T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S Table of Contents A B S T R A C T T A B L E O F C O N T E N T S L I S T O F F I G U R E S L I S T O F T A B L E S A C K N O W L E D G E M E N T S 1 I N T R O D U C T I O N 2 L I T E R A T U R E R E V I E W 2.1 T H E R M O D Y N A M I C S OF P A R T I A L L Y M I S C I B L E P O L Y M E R S Y S T E M S 2.1.1 Phase Stability and Criticality Conditions 2.1.2 Mechanisms of Phase Separation 2.2 R H E O L O G I C A L M E A S U R E M E N T S 2.3 T H E O R E T I C A L CONSIDERATIONS FOR M O D E L I N G R H E O L O G I C A L B E H A V I O R OF P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S 3 O B J E C T I V E S 4 R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P O L Y D I M E T H Y L S I L O X A N E / P O L Y E T H Y L M E T H Y L S I L O X A N E B L E N D S 4.1 INTRODUCTION 4.2 E X P E R I M E N T A L 4.2.1 Materials and Characterization 4.2.2 Methods -. 4.3 R E S U L T S A N D DISCUSSION 4.3.1 Viscoelasticity 4.3.2 Miscibility Analysis 4.3.3 Modeling of Phase Diagram 4.3.4 Effect ofMw ofPDMS on Interfacial Tension ofPDMS/PEMS Blends 4.3.5 Conclusion 5 R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P O L Y ( S T Y R E N E - C O - M A L E I C A N H Y D R I D E ) / P O L Y ( M E T H Y L M E T H A C R Y L A T E ) B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S I V 5.1 INTRODUCTION 63 5.2 E X P E R I M E N T A L 6 5 5.2.1 Materials 65 5.2.2 Methods 66 5.3 R E S U L T S A N D DISCUSSION 6 8 5.5.7 Thermal Properties 68 5.4 R H E O L O G Y A N D P H A S E B E H A V I O R 7 2 5.4. 1 Modeling the Phase Beha\>ior 84 5.4.2 Concentration Fluctuations 88 5.4.3 Conclusion 91 6 NON-LINEAR RHEOLOGICAL RESPONSE OF PHASE SEPARATED SMA/PMMA BLENDS 93 6.1 INTRODUCTION 93 6.2 E X P E R I M E N T A L 96 6.2.1 Materials: 96 6.2.2 Experimental Techniques 97 6.3 R E S U L T S A N D DISCUSSION 100 6.3.1 Non-Linear Viscoelasticity and Phase Separation 100 6.3.2 Effects of Preshear and Morphological Hysteresis 109 6.3.3 Thixotropic Loops 114 6.3.4 Transient Rheological Response 116 6.3.5 Dynamic Interfacial Tension 120 6.3.6 Conclusion 125 7 CONCLUSIONS 126 7.1 CONTRIBUTIONS TO K N O W L E D G E 130 7.2 R E C O M M E N D A T I O N S 131 REFERENCES 133 APPENDIX I - SLIDING PLATE RHEOMETER 142 APPENDIX n - P A R A L L E L PLATE RHEOMETER 144 APPENDIX III - CONE AND PLATE RHEOMETER 146 APPENDIX IV - CAPILLARY RHEOMETER 149 NOTATION 153 T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S V List of Figures Figure 2-1 The processing window for polymer blends typically lies in-between the glass-transition temperature (7/g) and the thermal decomposition temperature (Td) 6 Figure 2-2 Schematic diagram of two different polymer molecules on a two-dimensional lattice. The solid line represents polymer "1" and dashed line "polymer 2" 7 Figure 2-3 Gibbs free energy of mixing at temperature T as a function of composition. The mixture is miscible in region I and decomposes into two phases in region II and III [Schwahn (1994)] 11 Figure 2-4 Theoretical phase diagram (mean field) for a symmetric (N\=N2=N) binary mixture of linear homopolymers [Bates (1991)] 14 Figure 2-5 Time evolution of structure in phase-separating binary homopolymer mixtures. Nucleation and growth results when a homogenous mixture is thrust into the metastable region of the phase diagram. Spinodal decomposition occurs when a mixture is placed in a thermodynamically unstable state. The driving force behind coarsening in both cases is the minimization of interfacial energy through a reduction in interfacial area [Bates (1991)] 15 Figure 2-6 (a) Simple shear flow; (b) Uniaxial (simple) extension 17 Figure 4-1 Chemical structure of PDMS and PEMS 29 Figure 4-2 Superposed (a) G' and (b) G" for different PDMS at r r e p25°C. The symbols are same for both (a) and (b) 37 Figure 4-3 Master curve of PEMS84 at reference temperature of 25°C 38 Figure 4-4 Activation energy is given by the slopes of PDMS and PEMS lines 38 Figure 4-5. (a) Master curve of PEMS84 at the reference temperature of-120°C 39 Figure 4-5. (b) Temperature dependence of shift factor fitted with W L F equation 40 Figure 4-6 Time-temperature superposition of non-linear data for different grades of PDMS and PEMS. 41 Figure 4-7 M w dependence of zero shear rate viscosity of pure PDMS 42 Figure 4-8 Failure of TTS for PDMS162/PEMS84(90/10) in linear (a) and non-linear (b) regime 43 Figure 4-9 Microscopic photographs of PDMS87/PEMS84(90/10) and PDMS50/PEMS84(90/10) at 25°C. 45 Figure 4-10 Radius of droplet vs. molecular weight of various PDMSx/PEMS84(90/10) blends at 25°C. 46 Figure 4-11 Microscopic photographs of PDMS8/PEMS84(27/73) as the temperature is reduced from miscible to immiscible region (binodal temperature is equal to 100°C) 48 Figure 4-12 Shear stress as a function of temperature. The change in slope marks the binodal 49 Figure 4-13 (a) Phase diagram of PDMS8/PEMS84. (b) Temperature dependence of Flory's interaction parameter, % 52 T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .VI Figure 4-14. Graphical determination of spinodal temperature, 7", from the plot of inverse of the intensity of the interdiffusion process (1/1 vs. inverse temperature. The solid line intersects the 1/Taxis at the point I/I; 54 Figure 4-15 Dependence of Critical temperature of the UCST of PDMS/PEMS blends on the molecular weight of PDMS for different values of molecular weight of PEMS 55 Figure 4-16 Dependence of interfacial tension of PDMS/PEMS blends on the M n of individual components at 25°C. The line represents the fit of Equation (4-7) 58 Figure 4-17 Palierne model prediction of the elastic modulus data for (a) PDMS87/PEMS84(90/10) and (b) PDMS50/PEMS84 59 Figure 5-1 Chemical structure of SMA and PMMA 62 Figure 5-2a Characteristic DSC curves indicating one glass transition of homogeneous SMA8/PMMA blends. The blend composition of SMA/PMMA is mentioned near the corresponding curve. The determination of Tg and the width T^=T\-T2 from the endotherm is also schematically illustrated 69 Figure 5-2b The width of glass transition Tw as a function of composition for (•) SMA8/PMMA; (•) SMA14/PMMA; (•) SMA32/PMMA. Lines are drawn to guide the eye 70 Figure 5-2c Composition dependence of the glass transition of SMA/PMMA blends with varying amount of MA. The continuous lines represent the fit with the Gordon-Taylor-Kwei equation (see Equation (5-1)). The fitted values of the parameters is mentioned in the figure 70 Figure 5-3a Storage modulus G' and (b) loss tangent tand of pure components viz. (o) SMA8; (Q) SMA 14; (A) SMA32; (V) PMMA obtained by performing dynamic temperature sweeps at 0.47 rad/s. The heating rate was 1.5 °C/min 73 Figure 5-4a Storage modulus G' and (b) loss tangent tand of pure PMMA obtained by performing dynamic temperature sweeps with an undried sample (o); a sample dried at 120°C for 24 hours (•); and a repeat on the previous sample after an interval of 5 minutes (A). The dynamic temperature sweeps were performed at 0.47 rad/s and 3% strain while the sample was heated at 1.5°C/min 75 Figure 5-5 Dynamic temperature ramps of the (a) storage modulus G' and (b) loss tangent (tan 8) for the SMA8/PMMA blend at different compositions, frequency co=0.47 rad/s and strain amplitude 1-2%. Lines are drawn to guide the eye. The symbols are the same for (a) and (b). The vertical arrow in (a) indicates the Theologically determined phase separation temperature (binodal) from the first cliange of slope, as the blend is heated with a rate of 1.5°C/min. 0 is the angle between the line with slope 2 and that with slope 1 (see text). The three zones in (b) correspond to the miscible region (zone 1), metastable region (zone 2) and phase separated region (zone 3) for SMA8/PMMA(25/75) blend 77 Figure 5-6 Dynamic temperature ramps of the (a) storage modulus G' and (b) loss tangent (tan 8) for the SMA14/PMMA blend at different compositions, frequency co=0.47 rad/s and strain amplitude 1-2%. Lines are drawn to guide the eye. The symbols are the same for (a) and (b). 6 is the angle between line with slope 2 and that with slope 1 (see text). The three zones in (b) correspond to die miscible region T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .VII (zone 1), metastable region (zone 2) and phase separated region (zone 3) for SMA14/PMMA(25/75) Figure 5-7 Dynamic temperature ramps of the (a) storage modulus and (b) loss tangent (tan 8) for the S M A 3 2 / P M M A blend at different compositions, frequency co=0.47 rad/s and strain amplitude 1-2%. Lines are drawn to guide the eye. The symbols are the same for (a) and (b). 8 is the angle between line with slope 2 and that with slope 1 (see text). The three zones in (b) correspond to the miscible region (zone 1), metastable region (zone 2) and phase separated region (zone 3) for SMA32/PMMA(25/75) blend, as the blend is heated with a rate of 1.5°C/inin 79 Figure 5-8 The compositional dependence of maximum of the loss tangent tan 8 for S M A / P M M A blends with various M A content. The maximum tanb was collected by measuring the height of peaks in tan8 in Figures 5-5b, 5-6b and 5-7b. Lines are drawn to guide the eye 80 Figure 5-9 (a) Characteristic master curves of G ' and G" for SMA14/PMMA(50/50) blend, showing a failure of time-temperature superposition principle above 250°C. The inset in (a) shows the activation energy plot (+) showing the change in activation energy at 250°C (indicated by the lines and arrow), (b) Cole-Cole plots for SMA14/PMMA(50/50) showing the occurrence of double peak beyond 245°C Figure 5-10 Rheologically determined quiescent phase diagram of (•) S M A 8 / P M M A ; (T) S M A 1 4 / P M M A ; (•) SMA32/PMMA. A l l closed symbols represent data points from dynamic temperature ramps and frequency sweeps The binodal temperatures for S M A 8 / P M M A were also determined by turbidity measurements (o). The lines represent the fit of Equation (4-5) to rheologically determined phase diagram of each blend: (—) S M A 8 / P M M A ; (—) S M A 1 4 / P M M A ; (—) SMA32/PMMA. The fitted values of parameter A and B are mentioned in Table 5-2 (see text).84 Figure 5-11 Temperature dependence of Flory-Huggins interaction parameter, %, for S M A 8 / P M M A (circles); S M A 1 4 / P M M A (squares); and S M A 3 2 / P M M A (triangles). Open symbols represent values of % calculated using Equation (3) and closed symbols represent those obtained from Equation (4). The continuous lines are calculated using x=A-B/T (see Table 2 for values of A and B) for (—) S M A 8 / P M M A ; ( - - ) SMA14/PMMA; (—-) SMA32 /PMMA 87 Figure 5-12 Enhanced concentration fluctuation as calculated using Equation (5) as a function of proximity to the phase separation temperature for (o) S M A 8 / P M M A ; (•) S M A 1 4 / P M M A ; (+) SMA32/PMMA. Lines are drawn to guide the eye 90 Figure 6-1 Phase diagrams of SMA32/PMMA for various shear rates ( • and solid line: no shear; A and dashed line: 10 s"1; • and dotted line: 100 s"1). Lines are drawn to guide the eye 96 Figure 6-2 (a) Flow curves for sliding plate and capillary measurements at various temperatures, (b) Corresponding master curves with horizontal shifting and reference temperature r r ei=215 0C in both cases 101 blend, as the blend is heated with a rate of 1.5°C/min. 78 (• 230°C; o 235°C; T 240°C; v 245°C; • 250°C; • 255°C; • 260°C) 82 T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S VI11 Figure 6-3 Activation energy plots for linear viscoelastic (small amplitude oscillatory shear, •), large amplitude oscillatory shear (•), sliding plate (•) and capillary (A) measurements 102 Figure 6-4 SEM of a sample of SMA32/PMMA(50/50) quenched to room temperature, treated under various conditions in the two phase region (220°C): (A) linear viscoelastic measurements (yo=0.005, co=0.6 rad/s); (B) Large Amplitude Oscillatory Shear in non-linear regime (yo=10, co=31 rad/s); (C) Capillary experiment at 100s'. Dark background regions represent the PMMA rich phase 104 Figure 6-5: (a) Large amplitude oscillatory shear (LAOS) response for SMA32/PMMA 50/50 in the single phase region at 210°C. (b) Respective LAOS data in the phase separated region at 220°C; t* in the time axis refers to 27i/f, so that the LAOS data for various frequencies are normalized and shown in a single plot 105 Figure 6-6 Superposed LAOS storage for SMA32/PMMA(50/50) (G\ a) and loss (G", b) moduli associated with the first harmonic of sliding plate experiments; strain amplitude was Yo=10 and data correspond to 6 cycles/measurement (•: 210°C; Q: 215°C; o: 220°C; A: 225°C). Inset: Arrhenius plot of shift factors for moduli data, indicating the phase separation temperature at 215 °C 108 Figure 6-7 Linear viscoelastic data (a: G'; b: G") for SMA32/PMMA 25/75 at 225°C with increasing preshear rate 110 Figure 6-8 Linear viscoelastic data (a: G'; b: G") for SMA32/PMMA 25/75 at 225°C with decreasing preshear rate I l l Figure 6-9 Dependence of zero shear rate viscosity (obtained from Figures 6-7 and 6-8) on shear strain and shear rate in the inset. Lines are drawn to guide the eye 113 Figure 6-10 Thixotropic loops performed with SMA32/PMMA 25/75 at 225°C and different shear rates (arrows indicate direction of shear) 115 Figure 6-11 First normal stress difference transient response at y =0.1 s"1 for SMA32/PMMA 50/50 (o) and its constituent phases from the phase diagram, SMA32/PMMA 25/75 (•) and SMA32/PMMA 60/40 (•), at 220°C in the phase separated region. The solid line represents the fit of Equation (6-7). 117 Figure 6-12(a) The variation in first normal stress difference with increasing quantity of SMA32 in SMA32/PMMA blend, at 210°C lying in one phase region 119 Figure 6-12(b) The variation in first normal stress difference with increasing quantity of SMA32 in SMA32/PMMA blend, at 230°C lying in 2 phase region 120 Figure 6-13 Shear stress transient response to start-up in simple shear flow for SMA32/PMMA 50/50 at 225°C (O) and 230 °C (V) and 1 s"1 along with the corresponding fits from Equation (2-21) 122 Figure 6-14 Shear stress transient response to start-up in simple shear flow for SMA32/PMMA 50/50 at 225°C (O) and 230 °C (v) and 10 s"1 along with the corresponding fits from Equation (2-21) 123 Figure 6-15 Shear stress transient response to start-up in simple shear flow for SMA32/PMMA 50/50 at 225°C (O) and 230°C (v) and 100 s"1 along with the corresponding fits from Equation (2-21) 123 Figure 6-16 Dependence of the ratio Oi/d on the shear rate in the non-linear region 124 THERMODYNAMICS AND RHEOLOGY OF PARTIALLY MISCIBLE POLYMER BLENDS ix Figure 1-1. Schematic diagram of the shear stress transducer 142 Figure II-l Parallel plate rheometer 144 Figure ffl-1 Cone and plate rheometer 146 Figure IV-1 Capillary rheometer 149 Figure FV-2. Bagley plot for determining the end correction 152 T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S X List of Tables Table 4-1. Properties of materials used 32 Table 4-2. Properties of the blends used 33 Table 4-3. Parameters used in the Palierne analysis of phase-separating PDMS/PEMS blend (25°C) 60 Table 5-1. Maleic Anhydride (MA) content in the S M A grades investigated 65 Table 5-2. Estimated values of parameters A and B for various 85 S M A / P M M A blends 85 Table 5-3. Dynamic asymmetry and pretransitional parameters 89 T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S xi Acknowledgements I am grateful to my supervisor, Prof. Savvas G. Hatzikiriakos, for his encouragement, skillful guidance, constructive criticism and academic freedom during the course of this study. The high standards set by him have not only greatly contributed to the quality of this work but also helped me become a better professional. I thank Prof. Marianna Kontopoulou for skillfully co-supervising my Ph.D. This work would not have been possible without her research support and useful suggestions. I also thank Dr. Dimitris Vlassopoulos for extending his excellent research insight and cooperation in the area of thermodynamics and rheology of polymer blends. I thank my parents, Sheel & Priya and brother, Nagesh for their inspiration, love and continuing support. Most of all, I thank my wife Nidhi who has been a source of strength and motivation for success. I hope that by now she has forgiven me for all the weekends and long hours spent away from home in the lab. I wish to thank Prof. Scott Parent, Queen's University and Prof. C A . Haynes, UBC for helpful discussions and exchange of ideas regarding thermodynamics and chemistry of SMA/PMMA blends. I thank my friends Alfonsius Budi Ariawan, Manish Seth and Eugene Rosenbaoum for their continuing friendship. Finally, I acknowledge Nova Chemicals, USA and Max-Planck Institute, Germany for generously donating the polymer samples. T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S 1 1 Introduction blending. This area of technology has drawn considerable attention from ixing of two or more homopolymers is commonly referred to as polymer polymer researchers all over the world for the latter half of the last century. Polymer blends represent more than 30% of all plastics sold today and are amongst the most rapidly growing areas of polymer technology. Some of the reasons for this increasing interest in polymer blends include rapid, inexpensive, tailor-made design of polymer materials with unique set of properties. For instance, polystyrene is commonly blended with rubber to improve its impact properties. The resulting high impact polystyrene (HIPS) blend is composed of two immiscible phases: rubber droplets in polystyrene matrix. On the other extreme, some homopolymers can be blended to make miscible blends i.e. blends, which are miscible on the molecular level at all conditions. However, few really miscible blends exist e.g. polyphenylene oxide/polystyrene. An intermediate category comprises of blends, which are partially miscible, i.e. miscible at the molecular level, only under certain temperatures and blend compositions. There is a great challenge in designing and producing partially miscible polymer blends by controlling their morphology, particularly during processing. The structure of blend constituents determines the final properties of the binary blend. Therefore, understanding the interplay of morphology, chemical structure and processing is essential. C H A P T E R 1 - I N T R O D U C T I O N T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .2 Partially miscible blends, displaying upper critical solution temperature (UCST) or lower critical solution temperature (LCST) behavior [Utracki (1990)] are of great interest, both from the scientific and the industrial points of view. The earliest experimental evidence of partial miscibility was reported by Bank et al. (1972). They found by differential scanning calorimetry (DSC) that solution-cast blends of polystyrene/polyvinylmethylether (PS/PVME) yielded a single, composition dependent glass transition temperature (Ts). However, heating the samples to 125°C induced phase separation. In recent years, the science of rheology has been successfully used to investigate the phase behavior and the concentration fluctuations near the phase separation temperature of partially miscible polymer systems [Han and Chuang (1985), Han and Yang (1987), Ajji and Prudhomme (1988), Stadler et al. (1988), Ajji et al. (1991), Mani et al. (1992)]. These authors analyzed the linear viscoelastic response at various temperatures. They observed that for temperatures in the homogeneous regime i.e. far from phase separation, the time-temperature superposition was applicable and a representation of the storage modulus (log G) versus the loss modulus (log G") resulted in a single master curve [Han et al. (1995), Nesarikar (1995), Vlassopoulos (1996)]. However, the blends exhibit thermorheologieal complexity (i.e. failure of Time-Temperature Superposition principle), when phase separation occurs. Despite these developments, and although it is clear that phase separation is responsible for the thermorheologieal complexity of polymer blends, the physical origin of this behavior is not fully understood yet. C H A P T E R 1 - I N T R O D U C T I O N T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .3 The applicability of partially miscible blends in real-life polymer material applications has been limited due to the lack of clear understanding of the dependence of their properties on temperature, composition and level of shear during processing. Although the viscoelastic response of immiscible polymer blends has been extensively and successfully modeled using the emulsion model proposed by Palierne (1990) [Minale and Moldenaers (1997), Graebling and Muller (1993), Vinckier et al. (1996), Lacroix et al. (1998), Jeon et al. (2000); see section 2.3], a systematic approach to connect the morphology to the rheological response in partially miscible polymer blends is scarce [Vinckier and Laun (1999)]. In this work, the above mentioned issues are addressed by conducting a systematic thermal and rheological study on the following two partially miscible polymer blends: (i) a model UCST blend of poly(dimethyl siloxane) and poly(ethyl methyl siloxane) abbreviated as PDMS/PEMS; and (ii) a LCST blend of poly(styrene-co-maleic anhydride) and poly(methyl methacrylate) with SMA component having 8%, 14% and 32%o by weight maleic anhydride content (abbreviated as SMAx/PMMA where the number x represents % MA content). C H A P T E R 1 - I N T R O D U C T I O N T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .4 2 Literature Review the growing and commercially significant area of selecting components for polymer-polymer blends. Thermodynamics provides a systematic way of nowledge of polymer thermodynamics and rheology is very important in analyzing polymer blends and developing an understanding of the effect of individual polymer structure on the morphology of the blends at various conditions. The physical properties of the blend depend on the morphology of the polymer mixture. Rheology is the science that deals with the way materials deform when forces are applied to them. The key words in this definition of rheology are deformation and force. Rheology is used as a tool to probe the morphology of polymers. This chapter discusses the thermodynamics of polymer-polymer miscibility along with the basic principles of polymer rheology. Additional literature review on individual blends is also included in various chapters. The working principles of various rheological tools used in this work can be found in Appendix I to IV. 2.1 Thermodynamics of Partially Miscible Polymer Systems Polymer blends can be characterized as 'miscible' or 'immiscible' with respect to their phase behavior. The term "miscibility of polymer blends" will be used for their dispersal at the molecular level. Although it is possible to blend two polymers by either melt-blending in an extruder or dissolving in a common solvent and removing the solvent, the procedure does not ensure that the two polymers will mix on a microscopic level. In fact most polymer blends are immiscible [Kumar and Gupta (1998), Utracki C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .5 (1990)]. This means that the blend does not behave as a single-phase material. It will, for example, have two different glass transition temperatures, which are representative of the two constituents, rather than a single Tg. Both miscible and immiscible polymer blends find application in the industry. A specific example in which immiscibility is beneficial is the impact modification of polystyrene by rubber. On the other hand, miscibility is important in applications where segregation of the constituents could lead to deleterious mechanical properties, such as might happen at a weld line in injection molding [Kumar and Gupta (1998)]. The phase behavior of polymer blends comprising amorphous polymers is experimentally well accessible in a "window" the upper bound of which is the thermal decomposition temperature of the polymer components and the lower one is the glass transition temperature of the system (Figure 2-1). In general, miscible blends display phase separation at elevated temperatures, as shown schematically in Figure 2-1, i.e. Lower Critical Solution Temperature (LCST) behavior can be seen. Usually, miscibility of polymers is coupled with ordering in the systems imposed by specific interactions. An increase in temperature weakens the specific interaction, which is equivalent to ascending disorder. Hence, an entropy-driven LCST occurs where the blend phase separates upon heating. Some miscible blends exhibit not only LCST behavior but also thermally induced phase separation upon cooling. Cooling causes a decrease in compressibility, which in turn is equivalent to enhanced repulsion between segments exceeding the specific interactions below an Upper Critical Solution Temperature (UCST). Thus, the repulsion between the segments turns out to be unfavorable for order or miscibility below a certain temperature and promotes phase separation. The C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .6 simultaneous occurrence of a LCST and an UCST in blends of high-molar-mass polymers is considered to be a general phenomenon [Kammer et al. (1989)]. But usually the UCST shifts far below the glass transition temperature and therefore, is not accessible experimentally. When the glass transition temperature is sufficiently low as in systems containing an elastomer as one of the components, the UCST may be confirmed experimentally besides a LCST [Ougizawa et al. (1985, Ougizawa and Inoue (1986), Saito et al. (1987), Cong et al. (1986)]. T L C S T U C S T 2 phases ——-1 phase 2 phases 0 1 • (vol. fraction) Figure 2-1 The processing window for polymer blends typically lies in-between the glass-transition temperature (Tg) and the thermal decomposition temperature (Td). The phase behavior of partially miscible polymer blends can be modeled using the classical Flory-Huggins theory [Flory (1953)]. This theory assumes that there is neither a change in volume nor change in enthalpy of mixing two polymers; the influence of non-athermal behavior is accounted for at a later stage. Thus, the calculation of the free C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S 7 energy change on mixing at a constant temperature and pressure reduces to a calculation of the change in entropy on mixing. This latter quantity is determined with the help of a lattice model using formulas from statistical thermodynamics. First, the total number of ways of arranging N\ segments of two polymers (7=1, 2), H , into «ceu lattice sites is calculated. Subsequently, the entropy of the blend is obtained by using Boltzmann's equation (s = k In 0.) [Flory (1953)]. Figure 2-2 shows a two-dimensional lattice being filled with two polymer chains where each chain is made up of a different polymer. Figure 2-2 Schematic diagram of two different polymer molecules on a two-dimensional lattice. The solid line represents polymer "1" and dashed line "polymer 2". The degree of polymerization, i.e. the number of repeat units that make up a polymer chain poses an effect on the thermodynamics of the polymer blend [Bates (1991)]. Most thermodynamic theories presume a single repeat unit volume, although in practice chemically different repeat units rarely occupy equal volumes. Therefore it is convenient to define a segment volume V corresponding to either of the repeat unit volumes (V\ or V2), or any suitable mean repeat unit volume. With this definition the C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .8 number of segments per polymer molecule is N\ = M n / pVN 0 where p and M„ are the polymer density and number average molecular weight, and /Vo is Avogadro's number. The total number of cells in the lattice is specified as «ceu. Based on this convention <j>; = N\lnce\\ where <J>i and fa are lattice volume fractions occupied by polymers 1 and 2 and N-, is the degree of polymerization of polymer /. The thermodynamic properties of a blend can be evaluated by using the Gibbs free energy of mixing [Schwahn (1994), De Gennes (1979)] which is the difference of the Gibbs free energy of the blend Gu and of the pure components G\ and Gj weighted by their volume fractions (J>i and (l-(f>i), respectively. A G M is a function of temperature 7" and volume fraction §\ of the polymer "1". A G M is given by where AHM is the enthalpy of mixing and ASu is the entropy of mixing. A G M must be negative for a miscible system as shown in Figure 2-3. For polymer blends with specific interactions the random mixing assumption for the entropy of mixing is no longer valid. According to this assumption the contribution to the free energy arising from any correlation's of the chain connectivity is ignored. In the A G M = GM - [<J>iGi+ (l-<t>i)G2 ] (2-1) AGM(7; fa)=AHU(T, fa)- TASM(T, fa) (2-2) C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .9 case, where specific interactions are present, the overall entropy of mixing, ASu is usually written as: where ASc is the combinatorial entropy of mixing for a random orientation. The non-combinatorial entropy of mixing A S N C is a correction for cases where some local orientation is important, as in the cases of blends with specific interactions. This orientation decreases the overall entropy of mixing although empirical parameters are often used to describe this effect. Nearly, 50 years ago Flory [Flory (1953)] and Huggins independently estimated the change in free energy per segment A G M associated with mixing random walk (Gaussian) polymer chains on an incompressible (<j>i+<j)2 = 1) lattice (shown in Figure 2-A S M = A S C + A S N C (2-3) 2), AGM =AH„-TUS„ =kT - ^ - l n ^ + - ^ - l n * 2 + M 2 x (2-4) J where and J C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S 10 where k is the Boltzmann constant, Txs the absolute temperature and N-, is the number of segments of component /' per polymer molecule (the degree of polymerization), AHM and A ^ M are the enthalpy and entropy of mixing respectively. The first two terms on the right hand side in Equation (2-4) account for the combinatorial entropy of mixing ASc. Because mixing increases the systems randomness, it naturally increases AiSc and thereby decreases the free energy of mixing. Large chains can assume fewer mixed configurations than small chains so that ASc decreases with increasing N,. The third term represents the enthalpy of mixing A H M and can either decrease or increase A G M . It represents the excess Gibbs energy relative to the combinatorial reference state. The enthalpic term used in the Flory-Huggins lattice treatment of polymer blends comprises a temperature dependent interaction parameter, %, which accounts for the interaction energy per polymer molecule. The % parameter is usually assumed to be a composite term that includes contributions from dispersive forces, specific interactions, non-combinatorial entropy effects and to a lesser extent, compressibility effects. The choice of a particular pair of monomers establishes the sign and the magnitude of the energy of mixing. The Flory-Huggins interaction parameter % [Bates (1991)], %=A+B/T, where T represents temperature, and A and B are empirical parameters accounting for the enthalpic term. 2.1.1 Phase Stability and Criticality Conditions For the discussion of phase stability let's consider the Euler equation [Schwahn (1994)]: C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 11 A G m = <|>imAni + (l-fam)A[i2 (2-5) With A|ij = Lii-u.; 0, the chemical potentials of mixing per unit volume of polymer "1" and "2". A G m is the molar Gibbs energy of mixing, 4>im represents mole fraction of polymer "1" and u,j and u.j° are the chemical potentials of the component / in the mixed and pure state, respectively. The region of metastability is bounded by the condition, [d(AGM) i at>i ]Tj> = [ ^ ( A G M ) / at>i2 ]TJ> = o (2-6) o < T I II III II I Figure 2-3 Gibbs free energy of mixing at temperature T as a function of composition. The mixture is miscible in region I and decomposes into two phases in region II and III [Schwann (1994)]. In Figure 2-3, A G M is plotted versus §i for a partially miscible system. There are the regions I, II, and III which are the stable, metastable ( [ ^ ( A G M ) / d§\2 ]T ,P> 0 ) and unstable ( [ ^ ( A G M ) / dfa2 ]T,p< 0) ones, respectively. In region I all fluctuations in <J>! C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S 12 increase A G M . In region II fluctuations with small amplitudes are stable because of [ ( ^ ( A G M ) / ctyi2 ]J,P> 0 while fluctuations with sufficient large amplitude become unstable. Finally in region III all long wavelength fluctuations lead to a decrease of A G M because of [(^(AGM)/ ctyi2 ]T,P< 0. In Figure 2-3, A G M is plotted together with the Euler relation of Equation (2-5) as a curved line. The straight line represents an ideal mixture. According to the tangent rule the phase boundary of the two-phase region or binodal is determined by the intercepts of both curves. According to the equilibrium condition the chemical potentials of both components must be the same in both phases. In the decomposed state the straight line of the Euler equation gives A G M of the system, and its difference to A G M of the mixed state is the driving force of the decomposition process. The boundary between metastable and unstable region is defined by [t?(AGu)l d<i>i2 hr= 0 (2-7) and is called the spinodal. In Figure 2-4 a schematic phase diagram with T and § as axes is plotted. The stable, metastable and unstable regions are indicated as in Figure 2-3. Spinodal and binodal touch each other at the critical point, the only point where stable and unstable regions are linked together. The following must therefore hold for the critical point [ ^ ( A G M ) / aba2 \TJP= [ ^ ( A G M ) / at)!3 ]rr= 0 (2-8) C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S 13 For polymer blends described by Flory-Huggins theory, the critical point (4>c, Xc) is found to be, where fa is the critical composition, %c is the critical value of interaction parameter and Ni and N2 represent the degree of polymerization of polymer "1" and "2". 2.1.2 Mechanisms of Phase Separation There are two mechanisms of phase separation viz. nucleation and growth and spinodal decomposition. The phase separation mechanism can dictate the morphology of an article made from a polymer blend, depending on how and at what rate a material passes through the phase diagram [Wilkinson and Ryan (1995)]. The mechanisms are illustrated in Figure 2-4 and are discussed below. Figure 2-4 shows a typical phase diagram for a symmetric polymer mixture exhibiting a LCST. The solid line represents the equilibrium line separating the one phase regime from the two-phase regime. The dashed line represents the stability limit. In-between the equilibrium and stability lines, lies the metastable region. Inside the equilibrium (solid) curve, 2 phases exist with compositions (J)' A and fa'A. In the metastable region (such as B ' ) , phase separation occurs by a nucleation and growth mechanism, while an unstable mixture (such as A ' ) spontaneously demixes by spinodal decomposition. ) 2 (2-9) C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 14 Classical nucleation theory predicts that small droplets of a minority phase develop over time in a homogenous mixture that has been brought into the metastable region of the phase diagram (for example, from point B to B ' in Figure 2-4). O n e p h a s e i I i i I . I 0.0 0.2 0.4 0.6 0.8 1.0 volume fraction of component A Figure 2-4 Theoretical phase diagram (mean field) for a symmetric (Ni=N2=N) binary mixture of linear homopolymers [Bates (1991)]. As the temperature is increased the polymer blend may change its state from solid to liquid. Initially droplet growth proceeds by diffusion of material from the supersaturated continuum. However, once the composition of the supernatant reaches equilibrium ((j)"A in Figure 2-4), further increases in droplet size occur by droplet coalescence or Ostwald ripening; the latter refers to the growth of large droplets through the disappearance of smaller ones. Because of the extremely low diffusivity (TJiffusivity—AV2) and enormous C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S 15 viscosity (r\~N?A) of polymers, the second stage of growth can be extremely slow and may result in unusual particle-size distributions. In the metastable state, homogenous mixtures must overcome a free energy barrier in order to nucleate a new phase. In the thermodynamically unstable state there is no such barrier, and mixtures phase separate spontaneously (for example, from point A to A ' in Figure 2-4). This process which was first described by Cahn (1965) 36 years ago, is known as spinodal decomposition. It results in a disordered bicontinuous two-phase structure that is contrasted in Figure 2-5 with the morphology associated with the nucleation and growth mechanism. The initial size of the spinodal structure (see Figure 2-5) is controlled by the quench depth; deeper quenches produce finer structures. Almost immediately after the bicontinuous pattern begins to form, interfacial tension drives the system to reduce its surface area by increasing domain sizes of the phase-separated structure. Nucleation & Growth Figure 2-5 Time evolution of structure in phase-separating binary homopolymer mixtures. Nucleation and growth results when a homogenous mixture is thrust into the metastable region of the phase diagram. Spinodal decomposition occurs when a mixture is placed in a thermodynamically unstable state. The driving force behind coarsening in both cases is the minimization o f interfacial energy through a reduction in interfacial area [Bates (1991)] C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S 16 In symmetric critical mixtures coarsening does not disrupt the bicontinuous morphology that evolves through a universal, scale invariant form, as depicted in Figure 2-5. The intricate structures associated with spinodal decomposition lead to a variety of interesting materials applications. These include polymer-based membranes, controlled porous glasses, and certain metal and ceramic alloys. Linear homopolymer mixtures have become one of the most attractive systems for studying spinodal decomposition in recent years. Molecular architecture plays an important role in determining polymer-polymer phase behavior [Bates (1991)]. Binary homopolymer mixtures at equilibrium consist of either one or two phases (neglecting crystallization). In the event of phase separation, interfacial tension favors a reduction in surface area that leads to macroscopic segregation. A density gradient also favors this segregation. However polymer melts are extremely viscous so that phase-separated homopolymers rarely reach equilibrium morphology. Consequently, molecular architecture, which strongly influences polymer mobility, plays an important role in the evolution of phase morphology. Branching in particular disrupts the basic mechanism of polymer motion (known as reptation) and leads to significant increase in polymer viscosity. 2.2 Rheological Measurements To learn anything about the rheological properties of a material, one must either measure the deformation resulting from a given force or measure the force required to produce a given deformation. As a measure of force, one can use the stress, which is defined as the ratio of the applied force to the area it acts on. Deformation can be described in terms of strain or rate of strain. C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S 17 There are two basic flows used to characterize polymers: shear and shear-free flows. For these two types of flow, the components of the stress and rate of deformation tensors take on a distinct form. The laboratory procedure that most closely approximates simple shear is to place a thin layer of fluid between two flat plates, clamp one of the plates in place, and move the second plate at a constant velocity, u, as shown in Figure 2-6. (a) Wetted area, A Area of face = A F Figure 2-6 (a) Simple shear flow; (b) Uniaxial (simple) extension Under no-slip conditions, the shear strain and shear rate can be written as follows: (2-10) (2-11) The velocity field is given as: C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S 18 Vx = y(t)y V = Vz = 0 (2-12) The components of the rate of deformation tensor are: Jo = Y(0 fo 1 0^ 1 0 0 vo 0 oy (2-13) and the stress tensor components are of the form: X X 0> XX X X 0 "xy yy 0 0 (2-14) Likewise for simple extension shown in Figure 2-6b, the components of the strain rate tensor are, Y*=e(0: ^2 0 0 ^ 0 - 1 0 0 0 - 1 (2-15) where e(t) is the Hencky strain rate. The stress tensor components are of the form: C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 19 fx XX 0 0 ^ 0 T yy 0 (2-16) 0 0 Polymer melts are non-Newtonian fluids. This means that they do not obey Newton's law of viscosity, that is where r\ is the constant viscosity. The complexity of the structure of these liquids makes it possible for the structure to vary with the shear rate and this change in structure results in a viscosity change. The following empirical "power law" is often used to describe the dependence of viscosity on shear rate at high shear rates [Dealy and Wissbrun (1990)], where X. is a characteristic time of the material and r\o is the zero shear rate viscosity. The complex rheological behavior of polymers has two important practical consequences. First, no single rheological property gives a complete rheological characterization of the material, and, second, the measurement of a rheological property requires careful control. A very short review of the rheological instruments and techniques used for rheological measurements in this thesis are presented in Appendix I T = ryy (2-17) n = Tio M n-1 (2-18) C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S 20 to IV. Detailed description of the rheological measurements and equipment can be found in Dealy (1982) and Dealy and Wissbrun (1990). It has been recognized that the application of shear can lead to structural changes in complex fluids such as polymer blends [see for example Larson (1992)]. In principle, it can be thought that as a consequence of entropy reduction upon chain deformation due to flow, the phase diagram may be altered both qualitatively and quantitatively. Experimental evidence based on a combination of rheological and scattering experiments, suggests that both shear-induced mixing and shear-induced demixing (cloudiness) can occur in polymer blends [Chopra et al. (1998, 1999), Hobbie et al. (1994), Larson (1992), Takebe et al. (1989)], depending on the amount of shearing and the molecular characteristics of the polymers (molecular weights, glass transition temperature). The physical mechanism of the flow-induced mixing or demixing is apparently a suppression or enhancement of concentration fluctuations, respectively [Hobbie et al. (1994), Larson (1992)], due to shear. Since, polymer blend systems are viscous in comparison to small molecule binary mixture, the level of shear required to see changes in phase diagram is quite high. Chopra et al. (1998) reported a shift of less than 5°C in phase separation temperature at 10s"1 for various composition of SMA/PMMA blends. The shear-induced phase diagram is shown in Figure 6-1. Phase behavior (both LCST and UCST) is an equilibrium property and the use of rheological techniques to determine them are justified as long as they do not disturb the equilibrium. It has been shown by several researchers that the linear viscoelastic measurements do not alter the phase equilibrium significantly and can be used for detecting binodal temperatures [Ajji and Prudhomme (1988), Ajji et al. (1991), Chopra et C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S 21 al. (1998, 1999), Kapnistos et al. (1996a,b), Onuki (1997), Vlassopoulos (1996), Vlassopoulos et al. (1997), Vinckier and Laun (1999), Zhang et al. (2000)]. 2.3 Theoretical Considerations for Modeling Rheological Behavior of Partially Miscible Polymer Blends The miscibility of partially miscible polymer blends depends on their composition and system temperature. A reversible change in morphology is seen in partially miscible binary polymer blend, as its temperature is increased from single phase to phase separated region. This change in morphology can be detected by analyzing the alteration in the rheological response of the blend. A number of researchers have shown the efficacy of rheological measurements in assessing the morphology of partially miscible polymer blends [Vlassopoulos (1996), Kapnistos et al. (1996a), Chopra et al. (1998, 1999), Vinckier and Laun (1999), Zhang et al. (2000), Han and Yang (1987), Utracki (1990)]. This section describes the theoretical considerations for analyzing and modeling the rheological response of partially miscible blends. Partially miscible binary blends are thermorheologically simple in the single-phase region and therefore all theoretical considerations valid for homogenous polymers are relevant to them [Utracki (1990)]. However, in the phase separated region the composition of the two phases differs from that of pure immiscible systems. In the case, where droplet-matrix morphology exists, the droplets in immiscible blends are made up of one of the blend components and the total volume fraction of droplets is equal to the blend composition. This is not true in the case of partially miscible blends as the droplet composition is obtained from the binodal curve and the total volume fraction of droplets C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .22 should be calculated from levers rule. For instance, a blend with composition (<\>A) equals to 0.5 in Figure 2-4, splits into two phases with matrix composition (J)'A (equal to 0.2) and droplet § " A (equal to 0.8) in the phase separated regions. Assuming that the droplet composition is equal to <}>dr, the volume fraction of droplets is given by the below mentioned levers rule [Vlassopoulos (1996)]: <k = (4>' A -<|>A)/( <j)' A -<j>" A) (2-19) The linear viscoelastic properties of phase separated partially miscible blends can be described to a first approximation by using simple incompressible emulsion models considering a suspension of droplets of one coexisting phase in the matrix of other. One of such models is the Palierne viscoelastic model (Palierne, 1990) developed originally by Schowalter et al. (1968) for an emulsion of two immiscible Newtonian fluids. According to Palierne the droplet-matrix interface gives rise to an elastic contribution proportional to the ratio of interfacial tension over particle radius (a/R) [Palierne (1990), Graebling and Muller (1993)]. Numerous studies have demonstrated the success of the Palierne model in describing the complex, typically bimodal, terminal relaxation of polymer blends which are either completely immiscible or partially miscible undergoing phase separation reaching dynamic equilibrium [Graebling and Muller (1993), Vinckier et al. (1996), Lacroix et al. (1998), Vinckier and Laun (1999)]. The complex modulus, G*(co), of the two-phase mixture assuming uniform particle size can be written as: C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .23 G » = G ; .. l + 3<tvr7(co) (2-20) where H(co) is given by 4 0 f l G « ( f f l ) + G . ' ( a ' ] + [2G (to ) + 3G ; ( a ) ] \ 6 G ; (co ) + 19 G (co ) 1 4 I 7r) t 2 G " ( ( 0 } + 5 G (* >]+ [G - " ( » ) - G ; (co ) ] |l6 G ; (co ) + 19 G ; (co ) ] and the subscript m and /' refer to the matrix phase and dispersed phase (inclusions), respectively. The symbol a represents the interfacial tension between the matrix and the dispersed phase. R represents the mean radius of the droplet. The use of mean radius for nearly monodisperse blends has been suggested by numerous authors [Graebling and Muller (1993), Vinckier et al. (1996), Lacroix et al. (1998), Minale et al. (1997)] and is in particular justified for systems, where droplet growth with respect to time and temperature is insignificant. Although, it is important to model linear viscoelastic properties of polymer blends in order to develop understanding of quiescent morphology for polymer compatibility studies, it is insufficient for describing the morphology changes in the non-linear regime particularly during fast changes in flow condition i.e. transient conditions. The shear and normal stress transients during the deformation process can be modeled to a first approximation by combining the approach of Doi and Ohta (1991) with the affine deformation theory for single droplet behavior. In this model, the morphology of a blend is described by a scalar, which represents the specific interfacial area and a tensor, which is a measure of the anisotropy of the interface. The model consists of a constitutive C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .24 equation relating the stresses to the interfacial structure and kinetic equations for the evolution of the scalar and the interface tensor. The complete description of the model along with relevant equations can be found in Doi and Ohta (1991). Vinckier et al. (1997a) derived transient stress equations for immiscible blends of Newtonian or weakly viscoelastic fluids with a distinct disperse phase in a continuous matrix. They assumed that in such a system, whose viscosity can be approximated using the linear mixing rule, the droplets deform into fibrils under large increase in shear and remain symmetric about their long axis during deformation with initial diameter d. It is noted that this may not necessarily be fulfilled in blends with a highly elastic matrix material [Levitt et al. (1996)]. As soon as the droplet shape can be approximated by a cylinder and up to the point where break-up of the fibrils (or the shape transition mentioned above) occurs, the stresses are given by [Vinckier et al. (1997a)]: a(Y,Y,<0 = (£<Ml,)r 2 2y d^y2+4 (2-21) 2ya<j)i dr 2 2y dJy2+4 J (2-22) where y is the shear rate, y the total strain and <bdr the volume fraction of inclusions. It has been reported that stratified or fibrillar morphology is dominant at high shear rates or in elongational flows, whereas a globular structure is present at low shear C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .25 rates [Grizzuti et al. (2000)]. The non-linear steady state viscoelastic properties of concentrated binary polymer blends can be modeled to a first approximation using morphology based mixing rules such as those proposed by Frankel and Acrivos (1970) and Choi and Schowalter (1975). Although useful in practical applications, these relations have limited predictive capabilities and to this day, a successful theory describing the flow behavior of partially miscible polymer blends is not available. C H A P T E R 2 - L I T E R A T U R E R E V I E W T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S 26 3 Objectives he primary objective of this work is to investigate the relationship of chemical structure, thermodynamics and rheology in partially miscible polymer blends. The general objectives of the thesis are: • Investigate the rheological and thermal properties of blend components and compare them with those of the blends. • Determine the phase separation temperature by using rheological and optical methods in both the linear and non-linear regime. • Model the phase behavior of the blends using Flory's statistical mechanical model [Flory (1953)] assuming to a first approximation that the interaction parameter depends only on temperature, according to %=A-B/T. More specific objectives of the PDMS/PEMS work are as follows, • Investigate the effect of molecular weight of individual components on the following: • phase diagram of PDMS/PEMS blend. • activation energy of PDMS and PEMS and its blends • blend morphology • Model the linear viscoelastic properties of blends of PDMS/PEMS in the phase-separated region with the emulsion model of Palierne (1990). Estimate the interfacial tension by using the latter emulsion model and a combination of optical measurements and theoretical calculations. C H A P T E R 3 - O B J E C T I V E S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .27 • Investigate the quiescent phase separation mechanism with the help of an optical microscope. Specific objectives for SMA/PMMA work involve investigation of its thermal and rheological properties in order to determine the following: • Effect of maleic anhydride (MA) composition on the rheology and phase behavior of SMA/PMMA blends. • Dependence of the interaction parameters on the composition (% M A content) of SMA/PMMA blends and the description of the rheologically detected phase diagrams using Flory's lattice model. • Sensitivity of non-linear viscoelastic properties to phase behavior. • Feasibility of established emulsion models to describe the linear response using the phase diagram without fitting parameters. • Effect of pre-shear on the morphological and rheological response. • Evolution of shear and normal stresses and morphology in the phase-separated blend during startup of simple shear flow. Thesis organization The essential results of this thesis have been published or submitted for publication [see Chopra e/ al. (2000, 2001a, 2001b, 2002a, 2002b)]. However, additional work and results not included in these papers are presented here. Chapter 4 is based on the paper Rheological and Optical Study of Partially Miscible Polydimethylsiloxane/Polyethylmethylsiloxane (PDMS/PEMS) Blends Containing High Molecular Weight PEMS, which is under preparation. This chapter C H A P T E R 3 - O B J E C T I V E S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .28 discusses the rheological results of PDMS/PEMS blends made with high Mw PEMS and different M w PDMS. Modeling of phase diagram and application of Palierne model to the linear viscoelastic response of latter blend is also discussed. Chapter 5 is based on the paper Effect of Maleic Anhydride Content on the Rheology and Phase Behavior of Poly(styrene-co-maleic anhydride)/Poly(methyl methacrylate) Blends published in Rheol. Acta. [Chopra et al. (2001a, 2002a)]. It includes a study on the effect of M A content on the phase behavior of SMA/PMMA blends using a variety of thermal and rheological techniques. These techniques include the use of parallel-plate rheometer and DSC. Chapter 6 is based on the paper NonLinear Rheological Response of Phase Separating Polymer Blends: Poly(styrene-co-maleic anhydride)/Poly (methyl methacrylate) published in Journal of Rheology [Chopra et al. (2000)]. It deals with the study of non-linear behavior of SMA/PMMA blends containing 32% by weight maleic anhydride using capillary and sliding plate rheometry and applies models to describe the behavior. Finally, in Chapter 7, important conclusions are drawn, recommendations for future work are given, and the contributions made to general knowledge in this field are discussed. C H A P T E R 3 - O B J E C T I V E S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .29 4 Rheology and Thermodynamics of Polydimethylsiloxane / Polyethylmethylsiloxane Blends ^ ^ * ^ \ oth PDMS and PEMS polymers are in liquid form at room temperature and fllj are particularly noted for their stability at temperatures as high as 150°C. ^ ^ ^ ^ Owing to their transparency, the mixtures of these polymers are used as model systems for experimentation [Horiuchi et al. (1991), Theobald and Meier (1995), Enders et al. (1996), Fytas et al. (1996)]. They are used in a variety of applications including lubricants, water repellents, release agents and defoamers. The chemical structures of PDMS and PEMS are shown in Figure 4-1. C H , C 2 H 5 ( - S i — Polydimethyl siloxane Polyethylmethyl siloxane Figure 4-1 Chemical structure of P D M S and P E M S . The thermorheologieal properties of a mixture of monodisperse (M w = 4.3-162 kg/mol) polydimethylsiloxane (PDMS) and high molecular weight polyethyl methyl siloxane (PEMS), (M w = 83.8 kg/mol) are discussed in this chapter. C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .30 4.1 Introduction In the past 20 years, significant research has been done in order to control the properties of polymers by blending them with other polymers. Most of these blends are immiscible, however careful selection of the structure and molecular weight of individual blend components can result in partial miscibility under accessible conditions. In order to demonstrate this, a binary blend of nearly monodisperse poly(dimethyl siloxane) and poly(ethyl methyl siloxane), abbreviated as PDMS/PEMS was selected. This specific system was chosen because both components have extremely low glass transition temperatures ( T ^ P D M S ^ -125°C and 7g ,PEMs= -132°C), thus Tg is not interfering with the phase separation. In addition, due to the proximity of the Tg's of the two components, this model system is considered as "dynamically symmetric". Also, both components are marginally elastic, and therefore any elasticity in the phase-separated region should essentially arise from the interfacial tension [Momper (1989)]. Moreover, a reasonable understanding of the equilibrium phase behavior and dynamics of PDMS/PEMS blends has been acquired and reported in a series of recent publications [Horiuchi et al. (1991), Enders et al. (1996), Fytas et al. (1996), Meier et al. (1992,1993), Theobald and Meier (1995)]. Horiuchi et al. (1991) found monodisperse PDMS of molecular weight, M w = 80-1.15 x 103 kg/mol to be partially miscible with monodisperse PEMS o f M w = 4-9 kg/mol with the phase behavior exhibiting Upper Critical Solution Temperature. Similar UCST behavior was reported by Enders et al. (1996) with binary blends of PEMS (M w = 31.2 kg/mol, polydispersity = 2.67) and PDMS (M w = 10.4, 15.5, 18.1, 24 kg/mol, polydispersity ~ 2). Both these authors have shown that miscibility and concentration C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .31 fluctuations of PDMS/PEMS blends are strongly dependent upon molecular weight and polydispersity. Several researchers have analyzed the linear and non-linear viscoelastic response of various binary polymer blends using morphological models in order to delineate morphological information. Such morphological models include the viscoelastic emulsion model proposed by Palierne (1990) to describe the linear response of viscoelastic emulsions. Palierne model has been extensively and successfully used for immiscible polymer blends [Graebling and Muller (1993), Lacroix et al. (1998), Vinckier et al. (1996), Jeon et al. (2000)]. The emulsion models proposed by Choi and Schowalter (1975) and Frankel and Acrivos (1970) are amongst those used for modeling steady state rheological data [Grizzuti et al. (2000)]. However, a systematic use of these models in order to obtain critical morphological and interfacial information with respect to molecular weight in partially miscible polymer blends is scarce. In this work, the effect of blend composition and molecular weight of blend components on thermodynamics, rheology and morphology of PDMS/PEMS blends is examined C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .32 4.2 Experimental 4.2.1 Materials and Characterization The characteristics of the materials used are listed in Table 4-1. PDMS87 and PEMS84 samples were obtained from Max-Planck Institute, Germany and PDMS 162, PDMS50, PDMS8 and PDMS4 were purchased from Polymer Source Inc., Montreal, Canada. Table 4-1. Properties of materials used. Polymer (kg/mol) MJM„ r\0 (Pa.s) , 25°C H PEMS84 83.77 1.14 31 827 PDMS162 162 1.35 1200 1622 PDMS87 86.896 1.13 69 1040 PDMS50 49.5 ' 1.18 23 573 PDMS8 8.1 1.08 Not determined 100 PDMS4 4.3 1.23 Not determined 44 PDMS/PEMS blends were prepared by solution casting. The method comprised of blending the required weight fractions of both PDMS and PEMS in a common solvent, ft-hexane. Once the solutions were formed, w-hexane was removed by keeping the blends in a vacuum oven at 80°C for up to a week. The blends studied in this work are mentioned along with their respective viscosity ratios in Table 4-2. The blends were not exposed to temperatures above 150°C either during preparation or experimentation as PEMS is reported to undergo degradation above 150°C [Enders et al. (1996)]. C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S THERMODYNAMICS AND RHEOLOGY OF PARTIALLY MISCIBLE POLYMER BLENDS 33 Table 4-2, Properties of the blends used. Viscosity ratio Blend . _ (T|PDMSx/T|PEMSx) at iJ5°C PDMS162/PEMS84 38.7 PDMS87/PEMS84 2.23 PDMS50/PEMS84 0.73 PDMS4/PEMS84 0.0022 4.2.2 Methods Shear rheometry. A controlled stress rheometer with a controlled strain option (Viscotech from Reologica) was utilized in the parallel plate geometry (20 mm diameter, 0.5 mm sample thickness), with air convection temperature control (accuracy ±0.1 °C). Measurements were carried out under N2 atmosphere to prevent any absorption of moisture and/or degradation at high temperatures. The small amplitude oscillatory shear measurements performed included: (i) Stress and strain sweeps to determine the limits of linear viscoelastic response. (ii) Isothermal dynamic frequency sweeps from 0.1 rad/s to 188.5 rad/s at a given linear strain in order to investigate the linear viscoelastic material functions over the whole accessible frequency range. (iii) In addition to the above, steady shear sweeps to obtain flow curves and viscosity at different temperatures were also done. The steady state temperature sweeps to find binodal temperature comprised heating the blend to a temperature above the UCST and keeping it there for at least thirty minutes before gradually reducing the temperature by 2°C/min into the phase separated regime. CHAPTER 4 -RHEOLOGY AND THERMODYNAMICS OF PDMS/PEMS BLENDS T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .34 Optical microscopy: The optical observations were carried out on an optical microscope from Olympus, model BX5 1TF equipped with a Linkam SCC450 shearing hot stage. The polymer blend samples were inserted between two quartz parallel plates and squeezed to 100-200 microns. In order to find the coexistence (binodal) curve, the polymer blend samples were heated to the miscible region and kept for about half an hour. During this period, the samples were sometimes sheared to break down large air bubbles into smaller ones. At this time the samples were quenched at the rate of 2°C/min with five minutes stop at every 5°C change and photographs were taken. Light scattering. The spinodal temperature of PDMS8/PEMS84(27/73) was determined with dynamic light scattering measurements (photon correlation spectroscopy, PCS), which were carried out with an ALV-5000 full digital correlator over the time range 10"7-103 s. This instrument monitored the time autocorrelation function of the scattered laser intensity in the polarized (W: both incident and scattered beams were vertically polarized with respect to the scattering plane) geometry, Gwiqfy, here q = (47i«/A,)sin 9/2 is the magnitude of the scattering wave vector, n the refractive index of the medium (blend), A, = 488 nm the wavelength of laser radiation generated with a Spectra Physics model 2020 Ar+ laser (operating at 200 mW, single mode), and 9 the scattering angle. Under homodyne conditions, Gw(q,t) is related to the desired field autocorrelation function g(q,i) through the expression Gw(q,l) = 1 + J*\a g(q,t)\2, where J* is an instrumental factor, calculated by means of a standard solution, and a is the fraction of the mean C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .35 scattering intensity arising from concentration fluctuations, with decay times slower than about 10"7s [Brereton et al. (1987)]. The measurements started at the highest temperature in the homogeneous region, with subsequent cooling. Equilibration time was 60 min at each temperature and measurements were taken with sampling time of 30 min. The scattering intensity was recorded at an angle of 135° and the average value was plotted against temperature. C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S .36 4.3 Results and Discussion 4.3.1 Viscoelasticity The elastic and storage moduli of pure PDMS, PEMS and their blends were obtained using small amplitude oscillatory shear measurements in a parallel plate rheometer. Owing to the low molecular weight of the pure components and the blends, no crossover in G' and G" was found in dynamic frequency sweeps in all cases. Master curves for various grades of PDMS used were obtained by horizontally shifting the G' and G" curves at different temperatures to the reference temperature of 25°C. The superposed G' and G" are shown in Figure 4-2a and Figure 4-2b, respectively. The master curve for PEMS84 is shown in Figure 4-3. The extracted values of the activation energies from the linear viscoelastic measurements of PDMS and PEMS are independent of molecular weight and are equal to 15.2 kJ/mol and 20.1 kJ/mol, respectively as shown in Figure 4-4. Similar values of activation energy have been reported in the literature: Minale et al. (1997) reported a value of 13.9 kJ/mol for PDMS having a zero shear rate viscosity of 100 Pa.s at 23°C. Momper (1989) reported 15.5 kJ/mol for PDMS of molecular weight 23.6 kg/mol and 20.5 kJ/mol for PEMS of molecular weight 29.5 kg/mol. The average molecular weight between entanglements, Mt of PEMS84 was determined using the following relationship (Dealy and Wissbrun, 1990), o , where p is the density of PEMS84, 7" is the reference temperature, G N 0 N is the plateau modulus and R is the gas constant. C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 37 CL 106 105 io< 103 102 101 10° 10-1 10-2 10-3 it>« 10-5 10-6 •to-2 (a) P D M S 1Cr1 P D M S 1 6 2 slope=2 10° 10< co*aT, rad/s P D M S 8 7 P D M S 5 0 102 • 2 5 ° C • 3 0 ° C • 4 0 ° C • 5 8 ° C 1 • 8 0 ° C 0 2 5 ° C 4 0 ° C B 6 0 ° C 8 0 ° C © 2 5 ° C i V 4 0 ° C 103 co a. CD 10 5 10" 103 102 101 10° L- • 1CH slope = 1 Newtonian viscosity 10- 2 10-1 10° 101 co*aT, rad/s 10 2 103 Figure 4-2 Superposed (a) G ' and (b) G " for different P D M S at T r ef=25°C. The symbols are same for both (a) and (b). C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .38 105 f 10* r 1 0 3 r 102 r 1 0 1 r (0 CL 10° r CD 1&1 -ID 1&2 r 1 a 3 r 1a4 r 10-5 r 1 0 - 6 : 10"2 PEMS84 slope = 2 1a1 10° 10' w*aT, rad/s 10 2 • 25°C • 30°C • 40°C V 58°C 10 3 Figure 4-3 Mastercurve o f PEMS84 at reference temperature o f 25°C. • PDMS87 0 P E M S 8 4 A PDMS162 v P D M S 5 0 Q -j I I I I 1 I I 1 I L_J I I I I I I I 1 I I I I I I I I I I I I 2.8 2.9 3.0 3.1 3.2 3.3 3.4 1000/T, K"1 Figure 4-4 Activation energy is given by the slopes o f P D M S and P E M S lines. C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 39 The master curve of PEMS84 near the Te i.e. -120°C is shown in Figure 4-5a. The plateau modulus of PEMS84 is about 75000 Pa. After using this value of G°N and density of 0.965 g/ml, a value of 16 kg/mol was obtained for Me. The shift factors obtained in Figure 4-5a are plotted against temperature in Figure 4-5b. 10- 3 10- 2 10"1 co*aT, rad/s 1 0 2 1 0 3 Figure 4-5. (a) Master curve o f PEMS84 at the reference temperature of-120°C. The below mentioned Williams-Landel-Ferry (WLF) equation is reported to be useful in describing the temperature dependence of shift factors [Williams et al. (1955)]. l og (« r ) = -Cx(T~Tref) C2+(T-Tref) (4-1) C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .40 where aT represents the shift factor, C\ and C7 are empirical parameters related to the free volume, and T and TTef denote the temperature and reference temperature respectively. Equation (4-1) was fitted to the data shown in Figure 4-5b. An excellent fit to WLF equation was obtained for the temperature dependence of shift factors. 101 io-: 10-3 10-4 10-5 1 0 - 6 1 1 e 1 1 1 1 1 1 1 1 1 1 1 j 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r— (b) P E M S 8 4 -120 -100 -80 • f rom Expt . W L F fit c ,=7 .4309 , c 2 = 5 4 . 9 0 5 T r e ( = - 1 2 0 ° C -60 T, °C -40 -20 Figure 4-5. (b) Temperature dependence o f shift factor fitted with W L F equation. The flow curves of pure components were obtained at different temperatures using steady shear experiments in a parallel plate rheometer. A mastercurve for each component was obtained by horizontally moving each curve to the reference temperature of 25°C. The superposed curves are shown in Figure 4-6. The shift factors thus obtained were plotted against 1000/rin the inset of Figure 4-6. As expected, exactly'"the same values of activation energies were obtained for PDMS and PEMS as those using time-temperature superposition (TTS) in the linear viscoelastic regime. The rheological C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .41 response depends on morphology, which in the case of pure components is not affected by the level of shear. This is not true in the case of two-phase polymer mixtures, where the morphology is shear and temperature dependent [Chopra et al. (2000)]. T I I I I I r o P D M S 8 7 -o P E M S 8 4 A P D M S 5 0 V P D M S 1 6 2 • A 2.8 2.9 3.0 3.1 3.2 3.3 3.4 A 1000/T, K 1 : i i i 1 i i i : 10- 3 10- 2 10" 1 1 0 ° 10 1 1 0 2 1 0 3 10" Y*aT, s"1 Figure 4-6 Time-temperature superposition o f non-linear data for different grades o f P D M S and P E M S . The molecular weight, M w dependence of zero shear rate viscosity, T]o of pure PDMS at 25°C is shown in Figure 4-7. The zero shear rate viscosity, r)0 is approximately linearly proportional to Mw below the critical molecular weight, Mc of 20 kg/mol for PDMS. The latter value ofM c for PDMS is in reasonable agreement with that reported in the literature (Dealy and Wissbrun, 1990). Above Mc, the dependence becomes much steeper with the r\0 varying approximately with 3.57 power of M w . It is known that the C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .42 viscosity begins to be more and more dependent on shear rate above Mc [Dealy and Wissbrun (1990)]. TO 10 4 10 3 10 2 10 1 10° 10-' 10-2 10-3 P D M S 10° • 25°C slope = 1 slope = 3.57 M =20kg/mol 10 1 10 2 M w , kg/mol _ i i i i_ 10 3 Figure 4-7 M w dependence of zero shear rate viscosity of pure P D M S . Figure 4-8a shows a typical master curve obtained by superposition of the linear viscoelastic material functions for PDMS 162/PEMS84(90/10) at various temperatures to the reference temperature of 25°C. Failure of time-temperature superposition due to the secondary relaxation or shoulder in G' in the terminal regime can be observed in Figure 4-8a. This failure of TTS is expected because of the different morphologies encountered at different temperatures [Kapnistos et al. (1996a), Vlassopoulos (1996)]. The enhanced elasticity seen as a shoulder in G' in the terminal regime can be attributed to the two-C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .43 phase morphology of the blend [Vlassopoulos et al. (1997), Jeon et al. (2000), Palierne (1990)]. 1 0 5 co Q. CD CD 1 0 4 1 1 0 3 k 1 0 2 t-10 1 l 10° 1 a 1 (a) PDMS162/PEMS84(90 /10) G " slope = 1 G' slope = 2 1000/T, KT1 1Cr2 10-1 10° 10 1 w*a T , rad/s 1 1 1 1 — i — i — i r~| 1 10 2 1 0 2 1 0 1 CD QL 1 0 ° 10-1 T I i i i i r~ (b) P D M S 1 6 2 / P E M S 8 4 ( 9 0 / 1 0 ) T ( ° C ) , a T 26 JD 30 31 3i • 2 5 ° C , 1.0 • 4 0 ° C , 0.67 • 6 0 ° C , 0.48 • 8 0 ° C , 0.35 A 1 0 0 ° C , 0.25 • 1 2 0 ° C , 0.18 10 3 2.8 2.9 3.0 3.1 3.2 3.3 3.4 1000/T, KT1 —i 1—i—i i i i I , i i i i i i i i I i_ 10- 3 TO"2 10-y * a T , s"' Figure 4-8 Failure o f I T S for PDMS162/PEMS84(90/10) in linear (a) and non-linear (b) regime. C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 44 The shift factors obtained by horizontally shifting the G' and G" curves follow Arrhenius dependence on temperature as shown in the inset of Figure 7a. The activation energy of phase separated PDMS 162/PEMS84(90/10) is equal to 17.1 kJ/mol. The TTS of the non-linear stress data of PDMS 162/PEMS84(90/10) was obtained by horizontally moving the stress, such that it superposes at lower shear rates. This technique was used by Chopra et al. (2000) for finding out the shift factors in the non-linear regime of the LCST mixture of SMA/PMMA. The mastercurve thus obtained is shown in Figure 4-8b. The shift factors follow Arrhenius relationship as shown in the inset of Figure 4-8b. The value of the activation energy of PDMS162/PEMS84(90/10) in the non-linear regime is equal to 14.4 kJ/mol, which is less than the one obtained in the linear regime. This means that the blend's work of adhesion between bulk phases is higher in the linear regime compared to the non-linear regime [Vinckier et al. (1996), Chopra et al. (2000)]. 4.3.2 Miscibility Analysis It has been well established by light scattering, rheological and turbidity measurements that PDMS/PEMS blends are partially miscible [Horiuchi et al. (1991), Enders et al. (1996), Fytas et al. (1996), Meier et al. (1992,1993), Theobald and Meier (1995)]. The morphology of various PDMS/PEMS blends was probed in the optical microscope. PDMS 162/PEMS84, PDMS87/PEMS84, and PDMS50/PEMS84 blends showed droplet and matrix morphology in-between 25°C to 150°C as shown in Figure 4-9. All photographs relating to Figure 4-9 and 4-10 were taken at room temperature after C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S THERMODYNAMICS AND RHEOLOGY OF PARTIALLY MISCIBLE POLYMER BLENDS 45 keeping it in the stage for about one hour. Due to high viscosity of blends, it can take large time to reach equilibrium morphology and therefore these photographs correspond to quasi-equilibrium morphology. No phase transition was observed for PDMS 162/PEMS84, PDMS87/PEMS84, and PDMS507PEMS84 blends. On the other hand, single-phase behavior appeared to exist in the case of PDMS4/PEMS84 blend. PDMS87/PEMS84(90/10) 25°C 1096J1 PDMS50/PEMS84(90/10) 25°C . . 1. >r. 1 1062u Figure 4-9 Microscopic photographs o f PDMS87/PEMS84(90/10) and PDMS50/PEMS84(90/10) The PDMS8/PEMS84 blend also displayed droplet matrix morphology at room temperature. The size of the dispersed phase varied with the Mw of PDMS. Figure 4-10 shows the increase in droplet radius (measured using random sampling) with increase in M w of PDMS in PDMSx/PEMS84 blends (x = 8, 50, 87, 162) at 25°C. Larger droplet size means higher interfacial tension, which means that the interfacial tension of PDMSxx/PEMS84 increases with increase in Mw of PDMS. This effect has been discussed in detail in section 4.3.4 and shown in Figure 4-16. CHAPTER 4 -RHEOLOGY AND THERMODYNAMICS OF PDMS/PEMS BLENDS THERMODYNAMICS AND RHEOLOGY OF PARTIALLY MISCIBLE POLYMER BLENDS 46 A phase transition was detected for PDMS8/PEMS84 revealing a UCST behavior. The coexistence (binodal) curve of PDMS8/PEMS84 was established with the help of optical microscopy and rheology. The optical microscopy of PDMS8/PEMS84 showed that as the temperature of the sample is reduced from the miscible to the immiscible region, the origin of an interconnected disordered bicontinuous two-phase structure is observed as shown in Figure 4-11. This marks the onset of phase separation (binodal). 15 5 1 1 1 1 1 1 1 1 I 1 1 i i ] i i i i 1 i 1 1 | 1 1 1 1 | 1 1 I 1 | 1 I I I J T T T l - • PDMSxjc/PEMS84(90/10) - Line to guide eye --< --i - 4 i i I i i i i 1 i i i i 1 i i i i 1 i ' ' ' 1 ' ' ' • 1 • • • • 1 • I I 1 1 ' I I I 0 20 40 60 80 100 120 140 160 180 M w of PDMS, kg/mol Figure 4-10 Radius o f droplet vs. molecular weight of various PDMSx/PEMS84(90 /10) blends at 25°C. It can be clearly seen that the bicontinuous morphology begins to form at 100+10°C in the latter case. Similar co-continuous structures have been reported in the literature [Vinckier and Laun (1999), Zhang et al. (2000)]. The sort of phase separation mechanism seen in Figure 4-11 appears to be that of nucleation and growth after taking CHAPTER 4 -RHEOLOGY AND THERMODYNAMICS OF PDMS/PEMS BLENDS T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .47 the phase diagram (Figure 4-13a) into account. As the temperature of PDMS8/PEMS84(27/73) is reduced to the binodal point, nucleation and growth is seen and once the percolation threshold is reached, bi-continuous morphology develops. Bi-continuous morphology is formed at much lower temperatures than binodal and have characteristic fine-grained pattern exhibiting a characteristic length scale [Cahn (1965)]. In the first stage of nucleation and growth, the stress contribution of concentration fluctuations is of elastic origin [Vlassopoulos (1996)]; in the later stages the interfacial tension causes an extra elastic stress [Vinckier and Laun (1999)]. It has been shown previously that phase transitions can also be detected by rheological techniques [Vlassopoulos (1996), Kapnistos et al. (1996a,b), Chopra et al. (1998)]. Usually, these techniques involve measurements in the linear viscoelastic regime at very low frequency. Due to the low viscosity of the PDMS8/PEMS84 system, the torque resolution during linear viscoelastic measurements was very low (see Table 4-1). Therefore, steady state measurements were done in the parallel plate rheometer at shear rate of 0.48s"1 (high enough to give valid torque readings) and the stress was plotted as a function of temperature as shown in Figure 4-12. As the temperature of the blend sample is reduced from the miscible temperature region to below the UCST, a slight unusual increase in stress is observed. This enhanced stress originates from interface relaxation in phase separating blends and is determined by strength of concentration fluctuation and the interfacial area per unit volume [Kawasaki and Ohta (1986)]. Similar increase in shear stress was reported by Chopra et al. (1998) at the shear-sensitive phase separation temperature in the case of SMA/PMMA blends. C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S THERMODYNAMICS AND RHEOLOGY OF PARTIALLY MISCIBLE POLYMER BLENDS 48 1050n < > Figure 4-11 Microscopic photographs of PDMS8/PEMS84(27/73) as die temperature is reduced from miscible to immiscible region (binodal temperature is equal to 100±10°Q. The scale is same for all pictures. CHAPTER 4-RHEOLOGY AND THERMODYNAMICS OF PDMS/PEMS BLENDS T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 49 1 0 2 10 ' 10° 10-' - 1 — i — i — i — i — r -i—|—i—i—i—i—|—i—i—i—r P D M S 8 / P E M S 8 4 0 . 4 8 s ' • 10/90 • 27/73 " 50/50 _L _L_ 20 40 60 80 100 120 140 160 T, °C Figure 4-12 Shear stress as a function of temperature. The change in slope marks the binodal. There is a concern that the shearing involved in the steady state measurement may influence the phase behavior. Theoretical and experimental studies have shown that shear does not affect the critical temperature, when the shear rate is low enough [Onuki (1997), Horst and Wolf (1993), Vinckier and Laun (1999), Matsuzuka et al. (1997)]. The resulting morphology however can be affected by shear flow, even at low rate. Quiescent optical analysis coupled with steady shear rheology of other compositions of PDMS8/PEMS84 blend yielded the phase diagram shown in Figure 4-13a. 4.3.3 Modeling of Phase Diagram The phase behavior of the PDMS8/PEMS84 blend was modeled using Flory's statistical mechanical model [Flory (1953)]. The Flory-Huggins interaction parameter C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S _ 5 0 was assumed to be dependent only on temperature to a first approximation, according to yj=A-BIT. The interaction parameter, %, was estimated by using the Flory-Huggins theory, according to which, at equilibrium the following two equations arise from the equality of chemical potentials of component 1 and component 2 in both phases viz. u.i' " and u.2'=u.2", ln(4) t ) + ( l - r 1 / r 2 ) (c | ) 2 - $ 2 ) * = Vi — (4-2) = l n ( ( | ) 2 ' / c ] ) 2 " ) + ( l - / - 2 / r , ) ( r j , , ' - ( | ) , " ) * « 2 r 2 ^ ' r2 (<t>, - <t>, ) where, (j)j (/=1,2) are the volume fractions of the components. Equations (4-2) and (4-3) have been successfully used by Friedrich et al. (1996), Chopra et al. (2000) and a similar form was used by Hale and Bair (1997) to calculate the % parameter for binary polymer blends. The volume fractions were calculated from the weight fractions assuming a vanishing excess mixing volume. The r, 0=1,2) are the number of segments per chain, which were chosen according to the equation ri=NiVi(V\V2)~ , where N-, is the number average degree of polymerization (see Table 4-1) and V\ (/=1,2) are the monomer volumes. A value of 1.39 x 10"22 cm3 was selected for monomer volumes for both PDMS and PEMS [Meier et al. (1992)]. The compositions §'\ and (j)" at equilibrium were obtained from the experimentally detected binodal phase diagram in Figure 4-13a. The x parameter was calculated at different temperatures using both Equation (4-2) and (4-3) C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .51 and plotted in Figure 4-13b. In theory, both expressions of x should yield the same result but it is seldom true [Friedrich et al. (1996), Chopra et al. (2000)]. An average value of x was used as suggested by Kim and Burns (1987). The average value along with the standard deviation is shown in Figure 4-13b. The following expression was obtained for %• X = -3.34x1 IT3 + 5.59 IT(K) (4-4) The values of A and B in Equation (4-4) are very close to those reported earlier by Horiuchi et al. (1991) and Meier and Momper (1992). For the sake of comparison the dependence of x on temperature as reported by Horiuchi et al. (1991) is also plotted in Figure 4-13b. These authors used cloud point measurements to detect phase separation temperature of l owM w PEMS blended with highM w PDMS. Nevertheless, their values of X are comparable to the ones determined in this work. Moreover, the positive values of x at the temperatures investigated in Figure 4-13b confirms their finding that PDMS/PEMS blends have unfavorable enthalpy of mixing or no specific interaction, even for high M w PEMS. This suggests that the presence of UCST in PDMS/PEMS blends is driven by the unfavorable enthalpy change upon mixing. The binodal curve shown in Figure 4-13a was obtained by using temperature tie lines at 20°C, 80°C, 100°C and 130°C and Equation (4-2) and (4-3) as suggested by Kumar and Gupta (1998). Due to lack of data at 20°C and 130°C, the experimental binodal points were interpolated using a cubic spline. These interpolated values of <J>'PDMS8, <1>"PDMS8, <j>'pEMS84 and <()"PEMS84 obtained at 20°C, 80°C and 100°C were C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .52 substituted into Equation (4-2) and (4-3) to obtain two values of %. The average of these two values is plotted against temperature in Figure 13b. The error bars are large due to lack of experimental data. o 200 150 100 50 ' 1 1 1 1 1 1 ' ' 1 1 ' ' ' I • , ... I.. 1 , \ (a) P D M S 8 / P E M S 8 4 1 1 1 1 1 1 • Rheology -O Optical Microscopy T Light scattering . spinodal: x=5.59/T-3.34x10" 3 - Binodal from Flory Huggins / / o / \ / / A • / / \ \ 0 / / q \ -/ / A : / / \ \ • \ \ ' \\ ' / / / / \ \ • / % / 1 1 • • / \. ' 0.0 0.020 0.018 t 0.016 0.014 0.012 :-i 0.010 0.008 0.006 0.004 0.002 0.000 0.2 0.4 0.6 0.8 •PDMS (b) P D M S 8 / P E M S 8 4 • P D M S 8 / P E M S 8 4 — Z = 5.59/T-3.34x10" 3 Horiuchi etal., 1991 2.4 2.6 2.8 3.0 3.2 T"1 (X10 3), K"1 3.4 1.0 3.6 Figure 4-13 (a) Phase diagram of P D M S 8 / P E M S 8 4 . (b) Temperature dependence of Flory's interaction parameter, X-C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .53 The spinodal curve was predicted by using the following equation originally proposed by Flory and Huggins [Flory (1953)]: where fa is the weight fraction of component / and Ni represents the degree of polymerization of component /'. Here component "1" is PDMS8 and component "2" is PEMS84. The values of A and B can be obtained from Equation (4-4) and are equal to -3.34xl0"3 and -5.59K respectively. The spinodal curve generated by Equation (4-5) is shown by dashed line in Figure 4-13a. In order to confirm partial miscibility behavior of PDMS8/PEMS84, the spinodal temperature for PDMS8/PEMS84(27/73) was measured from dynamic light scattering technique [see Vlassopoulos et al. (1997), de Gennes (1979) and Brereton et al. (1987) for detailed information on light scattering data analysis]. As phase separation temperature is approached, the concentration fluctuations increase and the correlation length becomes larger. This results in an increase in intensity. At spinodal temperature, Ts the intensity is infinite (inverse of intensity, 1/1= 0). Thus, the spinodal temperature can be determined within the mean field theory [de Gennes (1979)] by extrapolating from homogenous region as shown in Figure 4-14. The intensity of this process was determined accurately, and the temperature at which it diverged was assigned to the spinodal point, Ts. This value of Ts is plotted in the phase diagram i.e. Figure 4-13(a) for comparison with the binodal points determined from rheology and optical measurements and Flory-Huggins theoretical predictions. As, T = 2 B (4-5) C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .54 expected the Ts for PDMS8/PEMS84(27/73) is lower than the binodal point. However, the predictions of Equation (4-5) are in poor agreement with the Ts. 0.4 4 0.3 -^ 0.2 -u . u - | , 1 , 1 , 1 , 1 1 0.0024 0.0026 0.0028 0.0030 0.0032 1/T(K"1) Figure 4-14. Graphical determination of spinodal temperature, T s from the plot o f inverse o f the intensity o f the interdiffusion process (1/7 vs. inverse temperature. The solid line intersects the 1 /T axis at the point 1/TS. The phase diagram of PDMS/PEMS blends is sensitive to the polydispersity and M w of individual components. In order to explain the dependence of critical temperature on Mw of PDMS and PEMS, Horiuchi et al. (1991) derived the following relationship using the Flory-Huggins theory: M, 2M wB 1 + \pAM wB pBM B wA (4-6) C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .55 where subscripts A and B indicate PDMS and PEMS, respectively and the density p is 0.975 g/ml for PDMS and 0.965 g/ml for PEMS. Mo represents the molecular weight of PEMS monomer unit and %m represents the interaction parameter corresponding to the critical point on the UCST as the blend components are nearly monodisperse. The critical temperature of the UCST for various polymer systems investigated by Fforiuchi et al. (1991) and Enders et al. (1996) is calculated by substituting the value of % from Equation (4-4) into Equation (4-6). The critical temperatures of PMDS/PEMS blends made with different Mw PEMS are plotted against Mw of PDMS in Figure 4-15. o 1000 800 600 400 200 10 1 Horiuchi et al., 1991 ( M w of PEMS=5.71 kg/mol) Enders era/ . , 1996 ( M w of PEMS=31.2kg/mol) This work Fits with x=5.59/T-3.34x10"3 10 2 10 3 M w of P D M S , k g / m o l 10" Figure 4-15 Dependence o f Critical temperature of the U C S T of P D M S / P E M S blends on the molecular weight of P D M S for different values of molecular weight o f P E M S . Equation (4-6) overpredicts the critical temperature data of Horiuchi et al. (1991) and Enders et al. (1996). Horiuchi et al. (1991) vised ternary mixtures of PDMS and bimodal molecular weight distribution PEMS and Enders et al. (1996) used blends made C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .56 with polydisperse PDMS and PEMS. According to Equation (4-6), one can clearly see that the predictions of rcrjticai (maxima on the UCST) range from 20°C to 770°C for the blends studied in this work (mentioned in Table 4-2). UCST can be detected only for PDMS8/PEMS84 in the temperature range investigated i.e. 25°C-150°C. The r^cai for the blend made with lowest M w of PDMS i.e. PDMS4/PEMS84, is predicted to be less than 25°C (see Figure 4-15). 4.3.4 Effect of M w of PDMS on Interfacial Tension of PDMS/PEMS Blends In PDMS/PEMS blends made with nearly monodisperse blend components, reducing the M w of PDMS, provided that the M w of PEMS is fixed, lowers the critical temperature of UCST. This can be clearly seen in Figure 4-15. Another evidence of growing miscibility with reduction i n M w of PDMS is given by the observed reduction in radius of the dispersed droplets (Figure 4-10). One would also expect to see a reduction in interfacial tension with lowering of M w of PDMS. The interfacial tension of PDMS/PEMS was estimated theoretically using Equation (4-7), that was obtained by using the extension of Ffelfand-Tagami theory by Broseta et al. (1990). a = bp0kT(x/6) 1/2 7t 12* 1 1 • + K N A N B J (4-7) In Equation (4-7), the statistical segment length (b), monomer density (po), absolute temperature (7), Flory-Huggins interaction parameter (x), the degree of polymerization of polymers (N\), where A and B represent PDMSx (x = 114,50,8,4) and PEMS84 respectively and Boltzmann constant (k) are used to calculate the interfacial C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .57 tension a. The latter equation was used by Jeon et al. (2000) to estimate the interfacial tension of a PB/PI blend. The value of % used was obtained from Equation (4-4). The other parameters used in the calculation of the interfacial tension of PDMS/PEMS blend are given by PO=(PPDMSPPEMS) 1 / 2 with pPDMs=0.009 A" 3 , PPEMS=0.008 A" 3 , £ =(6PDMS£PEMS) 1 / 2 with 6PDMS =£ PEMS =6.3 A (Flory, 1949). The values of N; for various grades of PDMS and PEMS are listed in Table 4-1. The prediction of a obtained using Equation (4-7) for various PDMS/PEMS 84 blends are plotted as a function of ./VPDMS in Figure 4-16. According to Equation (4-7), a droplets rapidly below NPDMS equal to 200, which is a sign of growing miscibility. Blends made with PDMS8 and PDMS4 correspond to N?Dus values less than 200 in Figure 4-16. The optical (Figure 4-10) and thermodynamic analysis (Figure 4-15) of these blends shows that they are more miscible than the others investigated in this work. Therefore, the predictions of a obtained using Helfand-Tagami theory are in good agreement with the conducted miscibility analysis. The Palierne emulsion model (1990) was also used in order to investigate the effect of change o f M w of PDMS on interfacial tension of the phase-separated blend. It is well known that the enhanced elastic contribution typically observed at low frequencies in phase separated blends can be analyzed by means of the Palierne viscoelastic model (1990) described in section 2.3. In order to make sure that the time scale of the experiment is much smaller than the time scale of the morphological changes, the Palierne model analysis in this work was carried out at room temperature, where the morphology was steady state. The morphology evolution is much slower at lower temperatures because it relies on diffusion process [Vinckier and Laun (1999)]. C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .58 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i i i j i i i i | i i i i | i i i i u -- H e l f a n d - T a g a m i theory -i O P a l i e r n e m o d e l i J i i i 1 i i i i ! t t i i 1 i i i i 1 i i i i 1 i i i i 1 i i i i 1 t i 1 1 0 200 400 600 800 1000 1200 1400 1600 N p D M S Figure 4-16 Dependence o f interfacial tension o f P D M S / P E M S blends on the M„ o f individual components at 25°C. The line represents the fit o f Equation (4-7). According to the theoretical predictions, based on Equations (4-4) and (4-6) and presented in Figure 4-15, the UCST of blends of PEMS84 with PDMS50 and PDMS87 should be at very high temperatures (above ~650°C). For the purposes of this work, the latter blends were considered as immiscible in the range of temperatures investigated. The storage moduli (G') of the droplet and matrix were used in the Palierne model (Equation 2-20) to fit the isothermal frequency sweep data using aJR as a fitting parameter. The experimental G' data for PDMS87/PEMS84(90/10) along with those of droplet and matrix are shown in Figure 4-17a. Similar fits for G' are shown for PDMS50/PEMS84 in Figure 4-17b. C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .59 1 0 " 1 0 3 1 0 2 1 0 ' co 1 0 ° Q_ b io-11 10 -2 1 0 - 3 t 1 0 - 4 10-5 1 0 " : (a! ,6> O P D M S 8 7 (matrix) V P E M S 8 4 (drop) • P D M S 8 7 / P E M S 8 4 ( 9 0 / 1 0 ) 4>=0.1 . a / R = 8 8 P a 10"' 1 0 ° co, rad/s 1 0 1 1 0 2 IO"2 IO"3 1 0 - 4 IO" 5 10"* 3 8C O P D M S 5 0 (matrix) V P E M S 8 4 (drop) • P D M S 5 0 / P E M S 8 4 ( 9 0 / 1 0 ) <t>=0.1, ot/R=135Pa 10- 2 10- 1 1 0 ° 10 1 1 0 2 co, rad/s 103 Figure 4-17 Palierne model prediction o f the elastic modulus data for (a) PDMS87/PEMS84(90/10) and (b) P D M S 5 0 / P E M S 8 4 . The Palierne model predictions of storage modulus for both PDMS87/PEMS84(90/10) and PDMS50/PEMS84 represented by a solid line in Figure 4-C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 60 17a and 4-17b agree reasonably well with the experimental data. The viscosity ratio of PDMS 162/PEMS84 is extremely high (see Table 4-2) and so the Palierne model is not applicable to this case [Palierene (1990), Vinckier et al. (1996)]. The resulting value of a/R can be found in Table 4-3. In order to measure the droplet radius, image analysis on the photograph of blends at various compositions (such as the ones shown in Figure 4-9) was done. The values of diameter have been summarized in Figure 4-10. The measured average diameter and calculated a for various blend compositions are listed in Table 4-3. The values of interfacial tension calculated using Equation (4-7) are compared with those estimated from Palierne model in Figure 4-16. It can be seen that good agreement is obtained between these two values. Table 4-3. Parameters used in the Palierne analysis o f phase-separating Blend Av. Diameter M a/ R (N/m2) a* (mN/m) PDMS87/PEMS84(90/10) 22 88 0.97 PDMS50/PEMS84(90/10) 14 135 0.95 The values of a* are given by (a/ R)x(Av. Diameter/2) 4.3.5 Conclusion Based on this work, it is concluded that PDMS/PEMS systems are partially miscible and their miscibility is very sensitive to the M w of individual components. More specific conclusions include: C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .61 • The activation energy of pure PDMS and PEMS is independent of molecular weight and level of shear for the range of molecular weights examined. However, the activation energy of blends is sensitive to blend composition and the level of shear. • In the phase-separated regime, the blends display droplet-matrix morphology, as evidenced by optical microscopy and rheological measurements. The radius of droplet and the interfacial tension increase with the increase in molecular weight of PDMS in PDMS/PEMS blends made with constant molecular weight PEMS. This trend confirms augmentation in immiscibility with increasing Mw of PDMS. • The emulsion model of Palierne was used to predict the secondary relaxation in elastic modulus using adjustable values of the ratio of interfacial tension over the droplet radius (oc/R). It was demonstrated that the predictions of interfacial tension obtained using Palierne model agree with Helfand-Tagami theory. • The results from optical and rheological methods corroborate with the application of the theoretical equation from Flory-Huggins theory to prove immiscibility and partial miscibility at the temperatures and compositions examined. The obtained value of interaction parameter suggests lack of specific interaction between PDMS and PEMS molecules. • TheM w of PDMS has to be below 10 kg/mol for the blends made with 83.8 kg/mol PEMS to show UCST below the degradation temperature i.e. 150°C. C H A P T E R 4 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F P D M S / P E M S B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 62_ 5 Rheology and Thermodynamics of Poly(styrene-co-maleic anhydride)/Poly(methyl methacrylate) Blends containing varying amount of maleic anhydride (MA) content, viz. 8%, 14% and 32% by weight. SMA/PMMA blends are industrially relevant mixture of a random copolymer of styrene and maleic anhydride (SMA) and poly methyl methacrylate (PMMA) displaying LCST behavior, and exhibiting high glass transition temperature and small dynamic asymmetry, i.e., difference in glass transition between components [Chopra et al. (1998, 2000, 2001a, 2002a)]. Although styrene is completely immiscible with PMMA [Utracki (1990)], it is known that the presence of maleic anhydride (MA) grafted to polystyrene affects both the physical properties of the resulting SMA copolymer and the thermodynamic and rheological properties of SMA/PMMA blends. The structure of SMA and PMMA is shown in Figure 5-1. hermorheological properties of partially miscible blend of poly(styrene-co-maleic anhydride) (SMA) and poly (methyl methacrylate) (PMMA) are discussed in this chapter. The different grades of SMA used include those f - CH2 — C H C H CH—} C O O C H 3 Polymethylmethacrylate Poly(styrene-co-maleic anhydride) Figure 5-1 Chemical structure of S M A and P M M A . C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .63 Blending of SMA and PMMA leads to an increase in motion rate of a phenyl group and a decrease in motion rate of a carbonyl group, which unambiguously indicates that there are strong interactions between the two groups [Feng et al. (1995)]. 5.1 Introduction The binary blend of poly(styrene-co-maleic anhydride)/poly(methyl methacrylate) is an industrially relevant mixture, displaying LCST behavior, and exhibiting high glass transition temperature and small dynamic asymmetry, i.e., difference in glass transition between components [Chopra et al. (1998, 2000)]. Although styrene is completely immiscible with PMMA [Utracki (1990)], it is known that the presence of maleic anhydride (MA) grafted to polystyrene affects both the physical properties of the resulting SMA copolymer and the thermodynamic and rheological properties of SMA/PMMA blends. Dedecker and Groeninckx (1999) reported that each extra wt% M A causes a T& increase of 2°C in SMA copolymer as determined by DSC. The phase behavior of blends involving several polymethacrylates and SMA containing varying amounts of MA was studied by Brannock et al. (1991) using cloud point measurements. Their results indicate that PMMA forms miscible blends with SMA containing M A in the range of 6% to 47% by weight. An assumption of repulsive forces within the copolymer leading to a net exothermic condition of blending was proposed to explain the miscibility of these blend systems. Feng et al. (1995) have shown that SMA/PMMA blends with 50wt% MA were miscible using a combination of nuclear magnetic resonance (NMR), Fourier transform infrared spectroscopy (FTIR) and DSC experiments. They suggested that a possible reason for the difference in their results with C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .64-Brannock et al. (1991), concerning the phase separation temperatures, is that the molecular weight of the SMA used by the latter authors was an order of magnitude higher than the one used in their study, thus affecting Flory's interaction parameter %. Their results indicated that SMA/PMMA blends are miscible on a molecular level and that there are strong intermolecular interactions between the phenyl groups in SMA and carbonyl groups in PMMA. It is the intermolecular interactions instead of the intramolecular repulsion forces within the SMA copolymer that make the SMA/PMMA blends miscible [Feng et al. (1995)]. Depending on temperature, both shear-induced mixing and demixing, typically at low and/or very high shear rates was observed in SMA/PMMA blends [Chopra et al. (1998); see also Aelmans et al. (1999)]. The shear-induced shift in phase diagram was modeled using a simple general thermodynamic model based on the established concept of generalized Gibbs energy of mixing with an extra entropic term due to flow [Wolf (1984), Horst and Wolf (1992), Chopra et al. (1999)]. In all of the previous work the molecular structure of the individual components was kept constant; the effect of molecular architecture of the individual components on the rheological properties of SMA/PMMA blends is virtually unexplored. The main focus of this chapter is to relate linear rheology of SMA/PMMA blends to their phase behavior, and in particular to detect Theologically the phase diagram for such systems, which have inherently low dynamic asymmetry. Since the dynamic asymmetry [Tanaka (1994)] is given by the difference in Tg of SMA and PMMA and the r g of SMA depends on its MA content, the dynamic asymmetry of SMA/PMMA blends depends on the amount of MA in SMA component. In this chapter, the following areas are investigated in detail: (a) the effect of M A content on the rheology and phase C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .65 behavior of SMA/PMMA blends and (b) the dependence of the interaction parameters on the composition (% MA content) of SMA/PMMA blends. A description of the rheologically detected phase diagrams using Flory's lattice model is also undertaken. 5.2 Experimental 5.2.1 Materials The random copolymers of styrene and maleic anhydride (SMA), containing 8% and 14% by weight maleic anhydride, were provided by Nova Chemicals, USA; another grade containing 32%MA was supplied by DSM, Netherlands. Poly(methyl methacrylate), PMMA, was provided by ICI, USA. The molecular characteristics of various grades of SMA and PMMA used in this study are listed in Table 5-1. Note the increase of Tg with increasing the M A content in SMA, in accordance with literature findings [Brannock et al. (1991)]. The polydispersity (MJMn) of all the polymers listed in Table 5-1 was about 2. Table 5-1. Maleic Anhydride (MA) content in the S M A grades investigated. Commercial Name Notation MA, %wt T & °C M w , g/mol Dylark 232 SMA8 8 125 200,000 Dylark332 SMA14 14 131 180,000 (research grade) SMA32 32 178 130,000 Perspex CP61-clear P M M A - 116 96,000 SMA/PMMA blends with variable MA content were prepared by solution casting. The method comprised blending the required weight fractions of both SMA and PMMA in a common solvent, acetone. Once the solutions were formed, acetone was removed by C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 66 keeping the blends in a vacuum oven at 80°C for a week. The obtained blend samples were stored in an oven at 120°C for at least 24 hours before usage. 5.2.2 Methods The effect of M A content on the rheology and thermodynamics of SMA/PMMA blends was investigated by using a combination of small amplitude oscillatory shear and DSC measurements on samples containing 8%, 14% and 32% M A content. Shear rheometry: A controlled stress rheometer with a constant strain option, Viscotech from Reologica, was utilized in the parallel plate geometry (20 mm diameter, 1 mm sample thickness), with air convection temperature control (accuracy ±0.1°C). Measurements were carried out under N2 atmosphere to prevent any adsorption of moisture and/or degradation at high temperatures. The small amplitude oscillatory shear measurements performed with each sample included: (i) Isochronal dynamic temperature ramps by increasing the temperature from the homogeneous to the phase separated regime, at a certain strain in the linear regime (2% to 5%) and heating rate (0.1°C/min to 2.5°C/min), in order to detect the onset of phase separation. Measurements were conducted at a frequency of 0.47 rad/s, which is sufficiently low to lie in the terminal regime [Vlassopoulos (1996), Chopra et al. (1998)], while at the same time providing satisfactory torque resolution; C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .67 (ii) Isothermal dynamic frequency sweeps from 0.1 rad/s to 500 rad/s at a given linear strain in order to investigate the linear viscoelastic material functions over the whole accessible frequency range across the phase diagram. Differential Scanning Calorimetry: A DSC-6 instrument from Perkin-Elmer was used for measurements of the glass transition temperature (Tg). Prior to the runs, a pre-heating step up to 200°C was performed to eliminate the thermal history of the samples, followed immediately by quenching to preserve the phase structure of the blend. The subsequent experiments were performed at a heating rate of 20 °C/min. The accuracy of the measurements is estimated to be +1°C. Turbidity Measurements: A simple set-up was constructed for performing transmission measurements in order to obtain the binodal temperatures independently. A Melles Griot He-Ne laser beam operating at 632.8 nm and 15 mW was sent through the sample blend and the transmitted intensity was detected in the forward direction using a photodiode detector (Newport 883-SL). The sample temperature was controlled by electrical heating (Eurotherm 810) using a platinum thermocouple and a home-made brass cell holder. The droplet in the intensity of the transmitted light by about 5-20% upon heating signified the phase separation C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .68 5.3 Results and Discussion 5.3.1 Thermal Properties A single Tg was detected in all thermograms, confirming that the blends are homogenous between room temperature and 200°C. Figure 5-2a shows thermograms of SMA8/PMMA blends. Similar thermograms were obtained for SMA14/PMMA and SMA32/PMMA blends. It is noted that the breadths of transitions shown in Figure 5-2a range from 13°C for pure PMMA and 9°C for pure SMA8 to 20°C for SMA8/PMMA(40/60). The breadth Tw here is defined as the difference between the temperature, T\, defined as the point of intersection of the tangent at T% with the extrapolated baseline of the liquid, and the temperature, 72, defined as the point of intersection of the tangent at Tg with the extrapolated baseline of the glass; see Figure 5-2a. Figure 5-2b shows the compositional dependence of the breadth of the transition Tw for each blend. Typically, there is an increase of the breadth of the glass transition relative to that of pure polymers. The maximum deviation occurs at a concentration of SMA, (J)SMA, equal to 0.4 to 0.5. The broadening of the glass transition in polymer blends is attributed to small-scale compositional fluctuations, causing nanoheterogeneities (Roovers and Toporowski, 1992; Roovers and Wang, 1994; Karatasos et al. 1998). Tw is also used as a measure of compatibility (Friedrich et al. 1996; Utracki, 1990). As a rule of thumb, for pure polymers one expects a 7^ of about 6°C, for miscible blends of about 10-20°C and for immiscible blends greater than 30°C. The Tw values obtained for all SMA/PMMA blends are well below 30°C (see Figure 5-2b) (Friedrich et al. 1996). If lower Tw is taken as a qualitative measure of compatibility then for the sake of C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .69 comparison, SMA14/PMMA appears to have higher miscibility than SMA8/PMMA or SMA32/PMMA at most compositions; such a remarkable dissymmetry in 7w of miscible polymer blends was reported before and also observed in dynamic mechanical measurements (Roovers and Wang, 1994). Figure 5-2c compares the T& as a function of composition for SMA/PMMA blends with varying amounts of M A content. i 1 — i — i — 1 — i 1 — i — i — | — i — i — i 1 — | — i — i — i — i 1 — i — i — i — i — [ — i 1 — i — i — | — i — i — i — r 60 80 100 120 140 160 180 200 T,°C Figure 5-2a Characteristic D S C curves indicating one glass transition o f homogeneous S M A 8 / P M M A blends. The blend composition of S M A / P M M A is mentioned near the corresponding curve. The determination of T g and the width T w = T i - T 2 from the endotherm is also schematically illustrated. C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .70 22 20 L (b) SMA Figure 5-2b The width of glass transition T w as a function of composition for (•) S M A 8 / P M M A ; (•) S M A 1 4 / P M M A ; (A) S M A 3 2 / P M M A . Lines are drawn to guide die eye. O 180 160 140 120 100 - i 1 1 r S M A / P M M A • 8 % M A O 14% M A T 32% M A k=1.077, q=4.62K k=1.16, q=0K k=1.545, q=1.34K 0.0 fSMA Figure 5-2c Composition dependence of the glass transition of S M A / P M M A blends widi varying amount o f M A . The continuous lines represent the fit with the Gordon-Taylor-Kwei equation (see Equation (5-1)). The fitted values of the parameters is mentioned in the figure. C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 71 The Gordon-Taylor-Kwei empirical equation, shown below, was found to provide a reasonably good fit to the data [see for example, Kwei (1984), Friedrich et al. (1996), Chopra et al. (1998)]: where, the subscripts 1 and 2 refer to the two components, fa, refers to the weight fraction of component /', q is a fitting parameter representing the extent of enthalpic interactions in the blend, and K « T&\IT%2- The quadratic term q§\fa is proportional to the number of specific interactions existent in the mixture. A rationalization of this term calls for the specific interactions (e.g. hydrogen bonds) that can be considered as effective cross-links, which increase the glass transition temperature by restricting the mobility of the polymer chains (Hale and Bair, 1997). Equation (5-1) was successfully used to describe the glass transition behavior of polymer mixtures in the past [Kwei (1984), Friedrich et al. (1996)]. The q values obtained after fitting Equation (5-1) to the data shown in Figure 5-2c are considerably lower than the value of 65K obtained in our previous work [Chopra et al. (1998)] involving melt blended SMA32/PMMA blends. Apparently, the improved droplet/matrix interaction due to the presence of solvent in solution blends is compensated by the formation of aggregates, which result in a less efficient network [Memon et al. (1998)]. Because of the formation of aggregates at higher compositions in solution cast samples, the interactions are somewhat reduced as indicated by the reduced amount of non-linearity in Figure 5-2c. This lower non-linearity C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .72 results in a lower value of the q parameter. Memon et al. (1998) reported lower elasticity in blends of PMMA and polybutylacrylate/styrene copolymer at higher blend compositions obtained by solution casting in comparison to those obtained by melt blending because of reduced interactions and formation of aggregates. They have confirmed this by SEM observations. 5.4 Rheology and Phase behavior Figure 5-3 shows dynamic temperature sweeps conducted on pure SMA (SMA8, SMA 14 and SMA32) and pure PMMA. The temperature sweeps were performed at 1.5°C/min, 0.47 rad/s and 1-2% strain. Since the maleic anhydride groups in SMA are prone to moisture absorption, which can transform them into acid groups by hydrolysis, pure SMA and SMA/PMMA blends were oven dried at 120°C for a minimum of 24 hours before use. Independent FTIR measurements were carried out to confirm that no hydrolysis of SMA took place. The dynamic temperature ramps on the oven dried samples of SMA8, SMA14 and SMA32 resulted in uniform decrease of elastic modulus with temperature. The abrupt decrease in storage modulus of SMA32 at about 242°C in Figure 5-3a can be linked to its degradation. This was corroborated by the sample appearance, which appeared slightly yellowish. However, this should not affect the phase separation temperatures of the blends, within the experimental accuracy of their determination (Chopra et al. 1998). The tanb corresponding to temperature sweeps on pure SMA and PMMA shown in Figure 5-3a are depicted in Figure 5-3b. There is a uniform increase in tan8 of the latter samples with temperature. C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .73 10 5 f TO Q. CD 10 4 U-103 U 102 10 1 10° " i r ~ i — i — i — i — i — i — i — i — i — i — r l i i I i i — i — i — i — i — i — i — r (a) J 1 I I I I I I I I I 1 I I I L _J I I I 1 I L. 180 200 220 240 260 280 300 T, °C 100 CO c to 10 n 1 1 r (b) - i 1 1 1 1 1 1 1 1 1 1 1 1 r 200 280 Figure 5-3a Storage modulus G ' and (b) loss tangent tanh of pure components viz. (o) S M A 8 ; (•) SMA14; (A) SMA32; (v) P M M A obtained by performing dynamic temperature sweeps at 0.47 rad/s. The heating rate was 1.5 °C/min. To illustrate the effect of moisture on PMMA, we performed dynamic temperature sweeps on dried and undried PMMA samples as shown in Figure 5-4. It was C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .74 found that a dynamic temperature sweep on undried PMMA resulted in an irreversible, abrupt increase in storage modulus accompanied with a distinct peak in tand at about 234°C. On the other hand, dried PMMA showed a reversible uniform decrease in the storage modulus with temperature. This is attributed to the presence of moisture in PMMA, which may initiate some kind of chemical gelation at 234°C as seen by a peak in tand in Figure 5-4. Also, the ceiling temperature of PMMA is about 220°C, which means that the gelation mechanism starts after the depolymerization of PMMA [Osborne et al. (1983)]. Dynamic temperature sweeps were performed on SMA/PMMA blends with varying M A content in the linear viscoelastic regime at a low frequency of 0.47 rad/s and a heating rate of 1.5°C/min. The temperature dependence of the storage moduli, G ' , for all blends is shown in Figures 5-5a, 5-6a and 5-7a. The change in slope of G ' relates to the binodal temperature [Kapnistos et al. (1996a, 1996b), Vlassopoulos et al. (1996, 1997), Chopra et al. (1998), Jeon et al. (2000)]. The temperature dependence of G ' in the phase separation region may be attributed to the composition fluctuations and the presence of a dispersed phase. The magnitude of the upturn in G ' is estimated by 6, where 6 = taw'^slope 2)-ta«"1 (slope 1), with (slope 1) and (slope 2) referring to the slopes of G ' before and after phase separation, respectively (see Figures 5-5a, 6a and 7a); even though this type of analysis may involve an error in the estimation of slope, it can be used for a qualitative comparison of the different blends. The values of 9 for critical compositions of SMA/PMMA blends with varying content of M A are 1.7° for SMA8/PMMA(40/60), 2.3° for SMA14/PMMA and 0.5° for SMA32/PMMA. In order to make sure that the observed change in the slope of storage modulus was because of phase C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 75 separation and not due to the gelation of PMMA, the blend samples were dried in the oven at 120°C for a minimum of 24 hrs and repeat runs were performed for each blend. 10° _J I L_ _l I I 1_ I I I I 200 220 240 T, °C 260 280 Figure 5-4a Storage modulus G ' and (b) loss tangent tanh of pure P M M A obtained by performing dynamic temperature sweeps with an undried sample (o); a sample dried at 120°C for 24 hours (•); and a repeat on the previous sample after an interval o f 5 minutes (A). The dynamic temperature sweeps were performed at 0.47 rad/s and 3% strain while the sample was heated at 1.5°C/min. C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 76 In this study it has also been observed that a peak in tand appears at the temperature where there is a change in slope of G ' . This is demonstrated in Figures 5-5b, 5-6b and 5-7b. Thus, it seems that both a change in the slope of storage modulus and the corresponding peak in tand can be used to detect the onset of phase separation (see for example Figures 5-5a and 5-5b). This observation is particularly useful in the case where the change in slope of G ' in a temperature sweep is subtle (for example in Figure 5-7a). A peak in tand typically represents an energy absorbing transition, where the area under the curve represents the amount of energy absorbed. It seems that the shape of the tand curves can be associated to the morphology of the SMA/PMMA blends. The pre-peak zone corresponds to a homogeneous structure (at least on a macro-scale) and represents the miscible region in the phase diagram. The peak zone in the tand curves (corresponding to the change in slope of G ' ) marks the onset of a transition phase, which may represent the metastable region in the phase diagram. Finally the post-peak zone appears to correspond to phase separated morphology where the blend consists of phase-separated domains. Figure 5-8 shows the magnitude of the tand peak near the point of phase separation (max. (tan 8)) as a function of SMA fraction for the three blends. It can be seen that SMA14/PMMA has the highest maximum tand followed by SMA8/PMMA and then by SMA32/PMMA. Considering that the area under the tand peak refers to the energy absorbed by the system to undergo phase separation, a higher value of peak in tand should correspond to higher energy absorption. More miscible systems, being more stable, should require more energy to phase separate. C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S THERMODYNAMICS AND RHEOLOGY OF PARTIALLY MISCIBLE POLYMER BLFNDS 77 Q_ CD 200 210 i I i i i i I i i i i I i i i i I (a) S M A 8 / P M M A • 25/75 40/60 A 50/50 v 60/40 • 75/25 220 240 260 280 T, °C Figure 5-5 Dynamic temperature ramps of the (a) storage modulus G ' and (b) loss tangent {tan 8) for the S M A 8 / P M M A blend at different compositions, frequency co=0.47 rad/s and strain amplitude 1-2%. Lines are drawn to guide the eye. The symbols are the same for (a) and (b). The vertical arrow in (a) indicates the rheologically determined phase separation temperature (binodal) from the first change o f slope, as the blend is heated with a rate of 1.5°C/min. <9is the angle between the line with slope 2 and that widi slope 1 (see text). The three zones in (b) correspond to die miscible region (zone 1), metastable region (zone 2) and phase separated region (zone 3) for S M A 8 / P M M A ( 2 5 / 7 5 ) blend. CHAPTER 5 - RHEOLOGY AND THERMODYNAMICS OF SMA/PMMA BLENDS T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 78 Figure 5-6 Dynamic temperature ramps o f the (a) storage modulus G ' and (b) loss tangent (tan 8) for the S M A 1 4 / P M M A blend at different compositions, frequency co=0.47 rad/s and strain amplitude 1-2%. Lines are drawn to guide the eye. The symbols are the same for (a) and (b). t9is the angle between line with slope 2 and that with slope 1 (see text). The three zones in (b) correspond to the miscible region (zone 1) , metastable region (zone 2) and phase separated region (zone 3) for S M A 1 4 / P M M A ( 2 5 / 7 5 ) blend, as the blend is heated with a rate of 1.5°C/min. C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .79 Figure 5-7 Dynamic temperature ramps of the (a) storage modulus and (b) loss tangent (tan 8) for die S M A 3 2 / P M M A blend at different compositions, frequency co=0.47 rad/s and strain amplitude 1-2%. Lines are drawn to guide the eye. The symbols are the same for (a) and (b). 6 is the angle between line with slope 2 and that widi slope 1 (see text). The three zones in (b) correspond to the miscible region (zone 1) , metastable region (zone 2) and phase separated region (zone 3) for S M A 3 2 / P M M A ( 2 5 / 7 5 ) blend, as the blend is heated with a rate of 1.5°C/min. C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 80 According to the latter criterion, SMA14/PMMA is more stable (thus less prone to phase separation) than SMA8/PMMA which, in turn is more stable than SMA32/PMMA. This agrees with the trend discussed earlier in the compatibility analysis of SMA/PMMA blends based on the breadth of glass transition shown in Figure 5-2b. A failure of time-temperature superposition principle (TTS) should also occur as the polymer mixture phase separates [Kapnistos et al. (1996a, 1996b), Vlassopoulos (1996), Chopra et al. (1998)]. This is confirmed in Figure 5-9a, for the G ' and G " master curves (obtained by horizontal fitting) of the various G ' and G " curves of SMA14/PMMA(50/50). Figure 5-8 The compositional dependence o f maximum of the loss tangent tan 5 for S M A / P M M A blends with various M A content. The maximum tanb was collected by measuring the height o f peaks in tanh in Figures 5-5b, 5-6b and 5-7b. Lines are drawn to guide the eye. C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .81 TTS failed at 250°C, which is slightly above the phase separation temperature of 245°C estimated by independent dynamic temperature sweeps of this composition. The temperature shift factors also showed slight deviation from Arrhenius behavior above 250°C (inset of Figure 5-9a). The occurrence of droplet/matrix morphology in phase separated SMA/PMMA blends is responsible for the deviation of temperature shift factors from the Arrhenius principle and the amount of deviation depends on the level of shear [Chopra al. (2000)]. The thermorheological complexity in the two-phase regime is due to the different morphologies encountered at different temperatures and the coarsening kinetics, which take place [Kapnistos et al. (1996b)]. The presence of two phases can also be demonstrated with Cole-Cole plots. These relate t|"(=G'/co) to t|'(=G'7(o) and are plotted in Figure 5-9b for SMA14/PMMA(50/50). The presence of two peaks in a Cole-Cole plot indicates that the polymer mixture has undergone phase separation [Dumoulin et al. (1991)]. Cole-Cole plots in the phase separated region display two frequency regions, corresponding to two different relaxation mechanisms; at high frequencies the relaxation phenomenon is essentially due to the phases' relaxation, whereas at low frequencies, the relaxation mainly stems from the deformability of the suspended droplets [Carreau et al. (1994)]. One can see the inception of a second peak at 245°C in Figure 5-9b, which grows in size with further increase in the temperature. It is noted that the phase separation temperature detected by dynamic temperature sweeps agrees very well with that obtained from a Cole-Cole plot. The same qualitative results are obtained when the linear viscoelastic data are analyzed in a Han plot representation [Kim et al. (1998)]. C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .82 CO Q. 2500 2000 h 1500 h 1000 500 h (b) S M A 1 4 / P M M A ( 5 0 / 5 0 ) O ° O O •o_ • O 0 1000 2000 3000 4000 5000 x\\ Pa*s 6000 7000 8000 Figure 5-9 (a) Characteristic master curves of G ' and G " for S M A 1 4 / P M M A ( 5 0 / 5 0 ) blend, showing a failure of time-temperature superposition principle above 250°C. The inset in (a) shows the activation energy plot (+) showing the change in activation energy at 250°C (indicated by the lines and arrow), (b) Cole-Cole plots for SMA14/PMMA(50 /50 ) showing the occurrence of double peak beyond 245°C (• 230°C; O 235°C; • 240°C; v 245°C; • 250°C; • 255°C; • 260°Q. C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .83 Both these techniques indicate that the phase separation in SMA14/PMMA(50/50) occurs at 245°C, in good agreement with the failure of TTS technique according to which, the phase separation temperature occurs in the range of 245°C to 250°C. By noting the change in slope of the storage modulus in Figures 5-5, 5-6 and 5-7, the rheologically detected phase separation temperatures for different blend compositions were collected, to yield the phase diagram depicted in Figure 5-10. According to Figure 5-10, SMA14/PMMA has the highest LCST, followed by SMA8/PMMA and then by SMA32/PMMA. The results are in excellent agreement with the predictions made by using the criteria of the breadth of the glass transition temperature (Figure 5-2b) and the magnitude of the max(ta«8) peak (Figure 5-8). The phase separation temperatures determined by turbidity measurements for SMA8/PMMA are also shown in Figure 5-10. The agreement between phase separation temperatures determined rheologically and from turbidity measurements is satisfactory over the whole range of compositions investigated. This confirms the demonstrated efficiency of rheological methods to determine the binodal temperatures of binary polymer blends, even for high T% systems like SMA8/PMMA [see also Vlassopoulos et al. (1996, 1997)]. The observed maximum miscibility at 14% MA content in SMA/PMMA corroborates the maximum miscibility range of 14%-33% M A content reported by Brannock et al. (1991). However, their cloud point values measured by using a hot plate are different from those reported in this work. The difference may be because of the different molecular weight and grade of the components used in both cases. C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .84 5.4.1 Modeling the Phase Behavior The phase behavior of the SMA/PMMA blends containing varying amounts of M A was modeled using Flory's statistical mechanical model [Flory (1953)]. By assuming that the Flory-Huggins interaction parameter, %, depends only on temperature, according to the relation %=A-B/T, the lattice treatment for mixtures of two polymers originally proposed by Flory and Huggins yields Equation (4-5) for spinodal temperature. The values of parameters A and B are obtained after fitting the Theologically determined phase diagram shown in Figure 5-10 with Equation (4-5). O o 255 250 245 240 235 230 "SMA Figure 5-10 Rheologically determined quiescent phase diagram of (•) S M A 8 / P M M A ; (T) S M A 1 4 / P M M A ; (•) S M A 3 2 / P M M A . Al l closed symbols represent data points from dynamic temperature ramps and frequency sweeps The binodal temperatures for S M A 8 / P M M A were also determined by turbidity measurements (o). The lines represent die fit of Equation (4-5) to Theologically determined phase diagram of each blend: (—) S M A 8 / P M M A ; (—) S M A 1 4 / P M M A ; (—) S M A 3 2 / P M M A . The fitted values of parameter A and B are mentioned in Table 5-2 (see text). C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .85 It must be noted that the rheologically detected phase separation temperatures shown in Figure 5-10 actually represents the binodal rather than the spinodal curves [Vlassopoulos et al. (1997)]. On the other hand, Equation (4-5) relates to the spinodal temperatures. However, since for high molecular weight systems, as the ones under consideration, the difference between binodal and spinodal curves is small [Manda (1998)], it is assumed that Equation (4-5) can be used reasonably for fitting the phase diagram. The values of A and B thus obtained are listed in Table 5-2. The values of A and B obtained in this paper are also compared with those obtained by fitting the phase diagram reported by Brannock et al. (1991) with Equation (4-5). The curves obtained by fitting Equation (4-5) to the experimental data are shown in Figure 5-10. Table 5-2. Estimated values of parameters A and B for various S M A / P M M A blends. Blend This work Brannock et aL (1991) A B ,K A B, K 0.24 122.1 0.14 64.3 0.12 60.2 0.17 96.3 0.14 69.0 Not studied The Flory-Huggins interaction parameter, % was estimated by two methods: (i) By assuming that x depends only on temperature, conforming to the relation %=A-BIT. The fitted values of A and B parameters mentioned in Table 5-2 were used to estimate the temperature dependence of X- This dependence is depicted in Figure 5-11. (ii) By using the Flory-Huggins theory, according to which, at equilibrium Equations (4-2) and (4-3) arise from the equality of chemical potentials of component 1 and component 2 in both phases viz. \x\' =\i\" and u 2 ' =p.2". The compositions §\ of the C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .86 components at equilibrium are obtained from the phase diagram (Figure 5-10). The % parameter is calculated at different temperatures using both Equation (4-2) and (4-3) and plotted in Figure 5-11. In theory, both expressions of % should yield the same result. From Figure 5-11, it can be seen that the calculated values of % are actually very close to each other. The values of % obtained by fitting the rheologically detected phase diagram (lines in Figure 5-11) are in reasonable agreement (within order of magnitude) with the ones obtained using thermodynamics [Equations (4-2) and (4-3)]. The slight difference between the values obtained by the two techniques may be due to the fact that Equation 2 predicts the phase diagram corresponding to the spinodal decomposition temperatures, whereas Equations (4-2) and (4-3) predict the phase diagram corresponding to the binodal decomposition temperatures. Nevertheless, Figure 5-11 can represent an indication of the effect of M A content on the interaction parameter. It can be seen that in phase separated SMA/PMMA blends, higher M A content leads generally to higher values of the interaction parameter confirming that blends containing 32% M A are less miscible. This means that higher M A content does not necessarily lead to higher miscibility but rather that there is an optimum M A content that corresponds to higher miscibility. In terms of simple thermodynamics for mixing of two components we can write [Flory (1953)]: A G M = A # M - T M M (5-2) C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .87 0.006 H 0.005 0.004 ~i 1 1 r / / / A A / A A * if A ~t i i i I 1 1 1 1-/ / / / - J i I I i I I I / i i ' i I 236 240 252 Figure 5-11 Temperature dependence of Flory-Huggins interaction parameter, %, for S M A 8 / P M M A (circles); S M A 1 4 / P M M A (squares); and S M A 3 2 / P M M A (triangles). Open symbols represent values of X calculated using Equation (3) and closed symbols represent those obtained from Equation (4). The continuous lines are calculated using %=A-B/T (see Table 2 for values o f A and B) for (—) S M A 8 / P M M A ; ( - -) S M A 1 4 / P M M A ; (—-) S M A 3 2 / P M M A . For blends to be miscible always AGM<0. Since ASM should always be positive for mixing (and its contribution is actually small), in order to have a miscible blend AHu should be as small as possible and ideally negative. For AHu we can also write: AHM-#AB"<j> A#A-<j>B#B (5-3) C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 88 where, HAB represents enthalpy of the blend and H represents the enthalpy of the individual component, where A=PMMA and B=SMA For our systems, as M A % increases beyond a certain composition, the system becomes less miscible, which means that AHu increases. This can be attributed to either an increase in HAB or a decrease in HB (the enthalpy of the pure component), or a relative variation of both. An increase in HAB essentially means is that there is an enthalpy penalty for combining the two components together, i.e. more energy needed to mix them. Therefore, in terms of enthalpic interactions, it seems that above a certain M A composition, there is less affinity for the components to interact with each other. This may imply in physical terms that there is stronger association within anhydride groups than between anhydride and PMMA. In SMA/PMMA blends, there is a competition between the inter (SMA) and intra (SMA-PMMA) molecular interactions. The present data suggest that the intra-molecular interactions dominate until 14% M A content in SMA/PMMA blends and the intermolecular interactions become more dominant above 14% M A content. 5.4.2 Concentration Fluctuations A measure of the extent of concentration fluctuations in the pretransitional region below the critical point is provided by the parameter e=2[l-(x/Xc)], where Xc is the value of x at the critical temperature Tc. This parameter becomes zero at the critical point, where the correlation length is infinite according to mean-field theory (see section 2.1). s is expected to depend on the dynamic asymmetry of the blend, which in turn depends on the M A content in the present case (see Table 5-3). C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 89 Table 5-3. Dynamic asymmetry and pretransitionalparameters. Blend composition (nearly critical) Dynamic asymmetry, °C (TpSMA-TgPMMA) T W , °C Phase separation temp., °C tan 8 e , ° de/ST, K-i SMA8/PMMA(40/60) 9 20.0 243 7.5 1.7 0.2 SMA14/PMMA(50/50) 15 14.5 245 7.4 2.3 0.1 SMA32/PMMA(50/50) 62 21.0 236 2.3 0.5 0.1 The parameter e has been used to describe the extent of concentration fluctuations in binary polymer blends and block copolymers and is given by [Onuki (1987), Kim et al. (1998)], p „ 4(1 - *2.e)b2,ENLN2 ( i - < |> 2 , c ) i + 4>2,cN2 l a r r 9 1 ^ (Te-T) ( 5-4) J where, Xc is the value of x at the critical temperature Tc, §2,c is the critical volume fraction of the reference component (PMMA component in all SMA/PMMA blends investigated), and A^ (i=l,2) is the degree of polymerization of component /. Figure 5-12 shows the dependence of s on the distance from the cloud point temperature for the SMA/PMMA blends. It is interesting that the slopes of the lines representing SMA14/PMMA and SMA32/PMMA in Figure 5-12 are virtually identical and smaller than the line representing SMA8/PMMA. This indicates that the enhancement of concentration fluctuations as the critical point is approached in the case of SMA8/PMMA is twice as strong compared to SMA14/PMMA and SMA32/PMMA (Table 5-3). In order to appreciate the effect of dynamic asymmetry on rheological and thermodynamic C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .90 properties, the critical blend compositions of SMA/PMMA blends with varying amount of M A content are analyzed in Table 5-3. The dynamic asymmetry of these blends with respect to the PMMA component increases with increasing amount of MA. Also as discussed earlier the Tw goes through a minimum and the LCST goes through a maximum as the amount of M A is increased in SMA/PMMA blends. The loss tangent (tan 8) decreases marginally as the MA content increases from 8% to 14% but thereafter it decreases significantly as the M A content approaches 32%. 6 h 4 h i i 1 r -1 1 1 r ~i 1 1 r -o- SMA8/PMMA _ • SMA14/PMMA -—•— SMA32/PMMA J* 0 0' .Br ,~-<a-" .0 . J2-_J I !_ 10 20 T -T, K 30 Figure 5-12 Enhanced concentration fluctuation as calculated using Equation (5) as a function of proximity to the phase separation temperature for (o) S M A 8 / P M M A ; (•) S M A 1 4 / P M M A ; (+) S M A 3 2 / P M M A . Lines are drawn to guide die eye. C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 91 It is interesting to note in Table 5-3 that 0, defined as the change in slope of elastic modulus in a temperature sweep upon phase separation signifying the magnitude of the upturn (Figs. 5-7), goes through a maximum at 14% M A content. The magnitude of upturn in G ' in a temperature sweep depends on the dynamic asymmetry of the blend [Kapnistos et al. (1996a, 1996b)]. In contrast to the latter behavior of G ' , the rate of increase of enhanced concentration fluctuations with temperature in the pretransitional regime reduces rapidly until 14% M A content after which the change becomes insignificant. But in the case of SMA/PMMA, where inter-molecular interactions appear to dominate the blend's dynamics above 14% MA content, the resulting decrease in chain mobility of SMA-rich domains in the vicinity of LCST may be responsible for the observed reduced rate of concentration fluctuations with temperature; as such, dynamic asymmetry is not the only relevant parameter influencing the pretransitional dynamics in interacting polymer blends. 5.4.3 Conclusion The application of a variety of thermal and rheological techniques allowed the thorough study of the phase behavior of SMA/PMMA binary blends as a function of M A content in the SMA copolymer. Miscibility in these blends was established by the presence of a single glass transition temperature (at temperatures below 200°C) exhibiting a small breadth, which was below 30°C. For comparative analysis, it was found that the miscibility of the blend increases with decreasing breadth of glass transition. Rheological detection of the phase separation temperatures (binodal) was based on the increase in elastic modulus and presence of a peak in tan5 during dynamic C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .92 temperature sweeps, appearance of dual peaks on the onset of phase separation in a Cole-Cole plot and failure of time-temperature superposition in dynamic frequency sweeps in the phase separated region. The magnitude of tan5 peak could provide a qualitative measure for comparing the miscibility of various blends, with higher miscibility being characterized by the higher peak in value in tanS. Experimental phase diagrams of SMA/PMMA using the above criteria exhibited a lower critical solution temperature with a nonmonotonic dependence on the M A content in the SMA copolymer, due to the interplay of interactions in SMA-SMA and SMA-PMMA, which affected the dynamic asymmetry and extent of pretransitional concentration fluctuations. The SMA copolymer containing 14% M A content yielded the higher miscibility in SMA/PMMA blends over all compositions. These phase diagrams were successfully described using the lattice treatment based on the Flory-Huggins theory and assuming only temperature dependence of the interaction parameter. In summary, the above methodological approach allows molecular control of the rheological and phase behavior of a variety of polymer blends, including industrially important systems which typically are characterized by high glass transition, high molecular weight, and varying degree of dynamic asymmetry. C H A P T E R 5 - R H E O L O G Y A N D T H E R M O D Y N A M I C S O F S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .93 6 Non-Linear Rheological Response of Phase Separated SMA/PMMA Blends he knowledge of thermal and linear rheological properties of SMA/PMMA blends developed in Chapter 5 is used in this chapter to understand the effect of processing conditions on morphology of a commercially important blend viz. SMA/PMMA with 32% MA content. This blend will be represented as SMA32/PMMA in this chapter. Various aspects of the blend's dynamic response are addressed by using parallel plate, sliding plate and capillary rheometry, and taking its phase behavior into account. The intent is to extend the theoretical concepts about SMA/PMMA blends developed so far to shear rates of industrial interest. It is demonstrated that the main rheological features of complex partially miscible, real industrial blends can be understood using relatively simple emulsion models, provided that their phase state is known precisely. 6.1 Introduction In real industrial blends, which typically involve high glass transition (Tg) constituents, the processing is usually carried out under non-linear conditions characterized by high values of strain and strain rate. To date, the work reported in the area of phase separated polymer blends in the non-linear regime is rather scarce, with the majority of the studies focusing on the simple shear flows obtained in parallel plate and cone-and-plate rheometers [Utracki, (1990), Sondergaard and Lyngaae-Jorgensen (1995)]. Significant advances have been made recently in the understanding of the C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .94 rheology-morphology relationships in fully immiscible systems. Key results include the demonstrated efficiency of rheo-optical techniques in probing the evolution of microstructure under flow, along with the indirect determination of the interfacial tension and the elucidation of the mechanisms of droplet break-up and coalescence in dense systems [Van Puyvelde et al. (1998), Vermant et al. (1998), Tsakalos et al. (1998)]; the successful use of the Doi-Ohta theory to describe the nonlinear response in steady conditions for mixtures of varying viscoelastic contrast [Vinckier et al, (1996), Takahashi et al, (1994)], as well as during start-up of simple shear [Vinckier et al. (1997a)]; and the analysis of stress relaxation and flow history to probe the microstructural changes with time [Vinckier et al. (1997b), Minale et al. (1997)]. Concerning the latter point, Minale et al. (1997) have reported a hysteresis loop in the volumetric mean radius while stepping up and down of the shear rate in a blend of low Te and nearly inelastic constituents, poly(dimethyl siloxane)/poly(isobutylene), PDMS/PEB. A hysteresis loop was observed below a critical shear rate, suggesting that the blend morphology depends on the initial conditions, and attributed to the hypothesis that stepping down the preshear rate results into droplet coalescence, whereas stepping up the preshear rate results into break-up. Based on rheological and optical observations, Takahashi and Noda (1995) concluded that the simultaneous occurrence of a minimum in shear stress and a maximum in normal stress in step-up shear experiments can be attributed to the break-up of the inclusions (dispersed phase). The shear and normal stress transients for PDMS/PEB, were successfully modeled by Vinckier et al. (1997a) by combining the theory of Doi and Ohta (1991), accounting for the viscoelasticity due to interfaces, and the affine deformation C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .95 theory for single droplet behavior. Their model was based on the assumption that the droplets initially deform into fibrils, which eventually break up into smaller droplets. As already mentioned, these developments relate to the dynamics of immiscible polymer blends consisting of low Tg nearly Newtonian components, and as such they are essentially ideal dense emulsions. Yet, the interplay of thermodynamics (i.e., slowing down of concentration fluctuations and time effects) and rheology, which represents a distinct feature of partially miscible polymer blends, remains an open area of research with not yet fully established phenomenology, both from the scientific and technological viewpoint, because of the ability to selectively tune the dynamic response of such systems. In addition, the shear rates generated in conventional rheometry are much smaller than those encountered in industrial operations. In continuation to chapter 5, the interplay of high shear rheology and thermodynamics in partially miscible SMA/PMMA polymer blends is discussed in detail. This blend is known to show both shear-induced mixing and shear-induced demixing, depending on the amount of energy stored to the system under flow [Chopra et al. (1998)]. The quiescent phase diagram along with the shear-induced changes is shown in Figure 6-1. This chapter discusses linear and nonlinear rheological response of SMA/PMMA and its interplay with the phase separated morphology. In particular the following aspects are addressed: (i) the sensitivity of nonlinear viscoelastic properties to the phase behavior; (ii) the feasibility of established emulsion models to describe the linear response using the phase diagram without fitting parameters; (iii) the effects of preshear on the morphological and rheological response; and (iv) the evolution of morphology in the phase separated blend during start-up of simple shear flow. C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S THERMODYNAMICS AND RHEOLOGY OF PARTIALLY MISCIBLE POLYMER BLENDS .96 Figure 6-1 Phase diagrams of S M A 3 2 / P M M A for various shear rates ( I and solid line: no shear ; A and dashed line: 10 s 1 ; • and dotted line: 100 s 1). Lines are drawn to guide the eye. 6.2 Experimental 6.2.1 Materials: The blend studied consists of a random copolymer of styrene and maleic anhydride (SMA), containing 32% by weight maleic anhydride, with Tg«175 °C, and a poly(methyl methacrylate) (PMMA) with 10 wt% copolymerized ethyl acrylate, 0.75 wt% of lubricating agent and 0.25 wt% of thermal and UV stabilizers, and with Tg«105°C. The weight average molecular weight of SMA is 130,000 and of PMMA 100,000, whereas both polymers have a typical polydispersity of 2. Owing to higher impurities (additives and other acrylates), the PMMA used in this chapter has lower Tg CHAPTER 6 - NON-LINEAR RHEOLOGICAL RESPONSE OF PHASE SEPARATED SMA/PMMA BLENDS T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S .97 than the one used in chapter 4. Before any use, the SMA was dried in a vacuum oven at 80°C for about 12 hours. SMA/PMMA blends of different weight compositions were prepared by melt blending using a twin screw extruder. The processed samples were stored in vacuum. The sample preparation procedure involved melt blending SMA32 and PMMA in a twin screw extruder. The melt temperature varied between 208°C-213°C, depending on the composition in order to avoid phase separation. For all compositions, a screw speed of 100 rpm was employed yielding an output of 3.63 kg/hr. During the various measurements at different temperatures discussed below, a certain test temperature was always approached starting from low temperatures, i.e., from the homogeneous region. 6.2.2 Experimental Techniques Rotational rheometric measurements: A Rheometric Scientific controlled strain rheometer (ARES-2KFRTN1) was utilized in the parallel plate (25 mm diameter, 1 mm sample thickness) and cone and plate geometries (25 mm diameter, 0.04 rads cone angle), with air convection temperature control (accuracy ±0.1 °C). The latter geometry was used to measure the first normal stress difference of the blends at a given temperature. Measurements were carried out under N2 atmosphere to prevent any adsorption of moisture and/or degradation at high temperatures. Even under these circumstances, degradation could occur after exposing the material for long times (typically more than 3 hours, although this depended on temperature) at high temperatures; it was accompanied by a change of viscoelastic properties and appearance of yellow of even brown color. These issues, which are discussed in detail by Chopra et al. (1998), were considered in C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 98 the present work, and thus degradation was not an issue whatsoever. The shear measurements included: (i) Steady preshear, followed by dynamic frequency sweeps. The latter were performed with both increasing and decreasing frequency. (ii) Thixotropic loops, where the shear stress or viscosity was recorded while the shear rate was slowly increased from zero until a specified value and then decreased back to zero with the same rate. (iii) Transient step shear rate experiments with the cone and plate geometry to measure first normal stress difference, N\. Non-Linear viscoelastic measurements: An Interlaken sliding plate rheometer with a 0.5 mm gap between the plates (of maximum length 12 cm) was used to perform Large Amplitude Oscillatory Shear (LAOS) as well as steady and transient shear experiments, at different temperatures (obtained using an air convection oven with an accuracy of ±0.5 °C). The sliding plate rheometer has a flush-mounted shear stress transducer [Dealy and Wissbrun (1990)] that measures the shear stress locally away from end effects; as a result, polymer degradation is not a problem. For LAOS experiments, each run consisted of a series of cycles having the same strain amplitude and covering a range of frequencies. There were 3-minute pauses between two consecutive frequency measurements. The test conditions were typically limited to the following ranges: (i) Strain amplitude kept constant at yo=10 units to be well into the non-linear flow regime (the value of yo=10 lies well within the non-linear regime, as indicated by C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S THERMODYNAMICS AND RHEOLOGY OF PARTIALLY MISCIBLE POLYMER BLENDS 99 comparing the 1st harmonic at different strain amplitudes with linear viscoelastic measurements). (ii) Frequency ranged from 0.01 to 5 Hz. For each combination of amplitude and frequency, the rheometer was programmed to produce 6-8 cycles. The data collection started on the 3rd cycle and continued over at least 3 cycles, resulting in 60 data points (20 per cycle). This was done in order to accommodate for the 20-point discrete Fourier transform method that was used to analyze the stress response into its harmonics. (iii) Start-up of stress in simple shear with transient times to reach steady shear ranging from 0.4 to 10 s, depending on the shear rate. Before starting an experiment at a given temperature, the sample was thermally equilibrated for a period ranging from 30 minutes to 2.5 hours (depending on the temperature) to ensure 'steady state' conditions and the same thermal history with the corresponding capillary experiments [Chopra et al. (1998)], which are compared below. CHAPTER 6 - NON-LINEAR RHEOLOGICAL RESPONSE OF PHASE SEPARATED SMA/PMMA BLENDS T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 6.3 Results and Discussion 100 6.3.1 Non-Linear Viscoelasticity and Phase Separation Flow curve Figure 6-2a represents the nominal steady shear flow curve for SMA/PMMA 50/50 obtained from sliding plate measurements. The corresponding capillary measurements reported previously by Chopra et al. (1998) are also included for comparison. According to Figure 6-1, the (quiescent) phase separation temperature for this blend composition is 215°C [Chopra et al. (1998)]; therefore, all data points in Figure 6-2a correspond to the phase separated region. Figure 6-2b shows the master curves for both sets of data, obtained by applying the principle of time-temperature superposition with reference temperature Tref=215°C; the latter was accomplished by shirting the stress data along the shear rate (horizontal) axis. It is noted that the capillary data are clearly higher than the sliding plate data. This is primarily due to the effect of pressure on the viscosity of polymer blends. This effect is described to a first order approximation by the empirical relationship [Dealy and Wissbrun (1990)]: n(P)=-n(0)exp(aoP) (6-1) where r|(P) is the viscosity at pressure P, r\(0) is the viscosity in the absence of pressure effects and aa is the coefficient of pressure dependence of viscosity; a value of the latter of 2.30 x 10"8 Pa"1 superposes the two sets of data satisfactorily. Such a value is in good agreement with those reported earlier for other molten (homo)polymers [Dealy and Wissbrun (1990)]. C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 101 10 3 102 co 10* t 10° (a) SMA/PMMA(50/50) Symbols Open: Sliding plate Closed: Capillary • • • • • o V o o T(°C) O 215 V • O • o 220 225 230 215 220 225 230 10 3 10 2 co 10 1 10° • • • 1 ' 1Cr1 10° ' 1 10' 10 2 10 3 Y. S- 1 .1 ' 1 1 1 1 1 1II 1 (b) SMA/PMMA(50/50) i i • i i i i n ) i T r e , = 2 1 5 ° C T • • • Symbols • 4 v Open: Sliding plate • Closed: Capillary : D T( °C ) o V O 215 V 220 : • • 225 O 230 : • 215 V T 220 • 225 • • 230 O I I I I , , | 10": 1Cr1 10° 10' 10 2 10 3 Y, S Figure 6-2 (a) Flow curves for sliding plate and capillary measurements at various temperatures, (b) Corresponding master curves with horizontal shifting and reference temperature Tref=2150C in both cases. The (horizontal) shift factors obtained from above procedure superposing the sliding plate and capillary measurements to produce master curves are plotted along with C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 102 those obtained from linear viscoelastic measurements [Chopra et al. (1998)] in Figure 6-3. 0.00198 0.00200 0.00202 0.00204 0.00206 1/T, K_ 1 Figure 6-3 Activation energy plots for linear viscoelastic (small amplitude oscillatory shear, • ) , large amplitude oscillatory shear (T), sliding plate (•) and capillary (A) measurements. The activation energy for non-linear measurements obtained from sliding plate and capillary rheometers are about the same, but significantly different from the activation energy in the linear regime, whereas the LAOS data lie in-between. It is also noted that from examination of the linear and nonlinear measurements it is evident that the Cox-Merz rule is valid only in the homogeneous region. Actually, during the linear viscoelastic measurements the polymer undergoes small deformation that disturbs the molecules very little from equilibrium. From the data of Figure 6-3, it is evident that the morphology evolution in the non-linear region is C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 103 different from that in the linear region, and this is reflected in a sharp difference in the activation energy. Actually, the value of the activation energy for the linear viscoelastic measurements is equal to 698 kJ/mol, whereas for the non-linear viscoelastic measurements it decreases to 499 kJ/mol (LAOS) and 303 kJ/mol; for comparison, the respective activation energies for SMA and PMMA in the linear regime are 297 kJ/mol and 161 kJ/mol. These values can be rationalized by considering the fact that the work of adhesion between the bulk phases SMA and PMMA (Wu, 1982) )^a=a.sMA+ctpMMA-c(. (where a is the blend's interfacial tension and the subscripts refer to the components surface tension), which relates to the excess elasticity due to interfaces [when the weighted component contribution is subtracted; see Vinogradov and Malkin (1980); Vinckier et al. (1996)] is higher in the linear regime compared to the non-linear regime. This would mean that for given conditions (temperature, rate) the blend's interfacial tension in the linear regime has to be lower with respect to the non-linear regime. The latter implies that a will decrease with increased curvature [Koberstein (1990)], suggesting that the shape of the (phase separated) inclusions in the linear region is more curved than those in the non-linear region. This is actually confirmed in the SEM pictures of Figure 6-4, where the SMA-rich domains are clearly elongated in the nonlinear region (Figure 6-4b) with respect to the linear case (Figure 6-4a); further, as the degree of nonlinearity increases (in capillary experiments), a fibriliar-type of morphology develops [Chopra et al. (1998)]. C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S -* 1.5 mm Figure 6-4 S E M of a sample o f SMA32/PMMA(50 /50) quenched to room temperature, treated under various conditions in the two phase region (220Q: (A) linear viscoelastic measurements (Yo=0.005, CO=0.6 rad/s); (B) Large Amplitude Oscillatory Shear in non-linear regime (Yo=10, co=31 rad/s); (Q Capillary experiment at 100s 1. Dark background regions represent the P M M A rich phase. Large Amplitude Oscillatory Shear response Figure 6-5a depicts typical cycles from a LAOS experiment for S M A / P M M A 50/50 at a temperature of 210°C. Under these conditions the blend is still in the single-phase regime. Maintaining the shear strain to a value of 10 units, into the non-linear regime, we can observe that the shear stress response in all cases is no longer truly sinusoidal, although periodic. Six to eight cycles were performed at each frequency in order to allow sufficient time for the evolution of the morphology at these operating conditions. The time of the LAOS experiments was scaled with reference to the data for the smallest frequency (0.05 Hz) for comparison purposes. It can be seen that a single peak was obtained in all these responses, similar to the response of homopolymers. Figure 6-5b depicts corresponding L A O S cycles (same conditions as in Figure 6-5a) of the same blend at 230°C, deeply into the phase separated regime (see Figure 6-1). CHAPTER 6 - NON-LINEAR RHEOLOGICAL RESPONSE OF PHASE SEPARATED SMA/PMMA BLENDS T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 105 300 200 -200 h -300 (a) SMA/PMMA(50 /50) y 0 = i o no. of cycles = 8 ^ \ 2 1 0 ° C / \ to, Hz 0.05 _ 0.5 / \ — 1 0.000 0.002 0.004 0.006 0.008 T/t* 80 40 to -40 -80 (b) SMA/PMMA(50 /50 ) no. of cycles = 8 to, Hz 0.000 0.002 0.004 0.006 0.008 T/t* Figure 6-5: (a) Large amplitude oscillatory shear (LAOS) response for S M A 3 2 / P M M A 50/50 in the single phase region at 210°C. (b) Respective L A O S data in the phase separated region at 220°Q t* in the time axis refers to 2%/f, so that the L A O S data for various frequencies are normalized and shown in a single plot. In this case, a double peak was unambiguously observed in all three responses that may be a signature of phase separation. Similar double-peak behavior was observed C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 106 for different SMA32/PMMA compositions in the two-phase regime. The occurrence of dual peaks is reminiscent of the analogous observation by Mani et al. (1992), who carried out step rate measurements in homogeneous polystyrene/poly (vinyl methyl ether) blends and assigned the second overshoot to shear-induced demixing; this feature could possibly serve as a sensitive tool for detecting transitions. However, the present situation is different, as the quenched sample retained its original phase separated morphology and exhibited shape of the phase separated domains which depended on the amount of shear deformation [Figure 6-4; Chopra et al. (1998)]. Actually, the balance of the viscous forces outside the domains and the Laplace pressure originating from the interfacial tension determines their final equilibrium form. In the case of SMA32/PMMA 50/50 the deformed inclusions (Figure 6-4) are composed of SMA32/PMMA 60/40 and owing to the higher content of SMA32 are more rigid than the matrix SMA32/PMMA 25/75 [Chopra et al. (1998,1999)]. It is conceivable that the form relaxation of these inclusions (which should be different from the matrix of different rigidity) relates to the double peak pattern of Figure 6-5. Dual relaxations were reported in linear viscoelastic measurements and related to form relaxation in systems of different rigidity, depending also on the amount of compatibilizer, the latter associated with the interfacial stress [Graebling et al. (1993), Kapnistos et al. (1996a), Rieman et al. (1997)]. Time-Temperature Superposition of LAOS data The most straightforward approach to analyze the LAOS data is to use Fourier transforms [Dealy and Wissbrun (1990), Larson (1988)]. The objective is to break the C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 107 stress signal into its components, i.e., obtain (1) the number of harmonics; and (2) their amplitude and frequency. For this purpose, a 20-point discrete Fourier transform (DFT) was employed. The DFT of time series o(nAt) is - ilnz n # „=<> N (6-2) where N is the number of data points, 20 in this case, / = V - 1 , Aco - 2nl(NAt), z is the discrete frequency component index ranging from 0 to A M . The sought values of the stress response are: zAco and or SN= —h arctan 2 SN = h arctan 2 o-„ =2\ad(zA(o\ n lm(ad (zA<y)) (6-3) Re(crd(zAe;)) Im(crd (zA<y)) Rt(ad(zAco)) ; for Re(<rd(zA<y)):> 0 ; for Re(o-d(zA<y))< 0 The same can be expressed in terms of storage and loss moduli of the individual stress harmonics: To and C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 108 The LAOS stress response was decomposed into its non-linear material functions (Gn', Gn") using DFT. Figures 6-6(a) and (b) depict the superposed (shift along the frequency axis only) non-linear material functions for SMA32/PMMA 50/50 corresponding to the first harmonic at the reference temperature of 7,ref=210oC. Figure 6-6 Superposed L A O S storage for SMA32/PMMA(50 /50 ) (G', a) and loss (G'\ b) moduli associated with the first harmonic of sliding plate experiments; strain amplitude was yo=10 and data correspond to 6 cycles/measurement (•: 210°C; • : 215°C; O: 220°C; • : 225°Q. Inset: Arrhenius plot o f shift factors for moduli data, indicating the phase separation temperature at 215 °C. C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 109 The failure of time-temperature superposition occurs at about 215°C. This critical temperature is in excellent agreement with the temperature obtained using small amplitude oscillatory shear and capillary measurements reported earlier by Chopra et al. (1998); thus within the rather limited range of temperatures examined, it seems that the sensitivity of nonlinear viscoelastic material functions to phase separation compares to that of their linear counterparts. The inset in Figure 6-6a shows the dependence of the (horizontal) shift factor, ax on \IT. The shift factors follow Arrhenius behavior both in the single and phase separated regions (see also Figure 6-3), although with different activation energies due to the morphological changes inside the phase diagram (see also Figure 6-4). In fact, it can be seen from the inset of Figure 6-6a that the activation energy in the homogeneous region (equal to 245kJ/mol) is clearly lower than that of the phase separated one (499kJ/mol), and is comparable in magnitude to that extracted from the pure components based on the additivity rule. 6.3.2 Effects of Preshear and Morphological Hysteresis Dynamic frequency sweeps following both increasing and decreasing pre-shear rates were performed for SMA32/PMMA 25/75 in the phase separated region (above 219°C, according to Chopra et al. (1998), under the same thermal treatment as outlined in the other experiments above. Figure 6-7 shows the experimental linear G' and G " with increasing pre-shear rate; the data were used to calculate the parsimonious spectrum for each dynamic frequency sweep curve [Baumgaertel and Winter (1989), Rosenbaum et al. (1998)], from which the corresponding zero-shear viscosities, r|o, were extracted. It may C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S THERMODYNAMICS AND RHEOLOGY OF PARTIALLY MISCIBLE POLYMER BLENDS 110 be noted that the values of the viscoelastic moduli exhibit a nonmonotonic dependence on the shear rate as they decay in the pre-shear rate range from 0.03 to 0.3 s"1, and increase from 0.3 to 1.0 s"1. 10* ID 103 -l i i i l 1 1—i T i i 111 i i—i i r i 11 - (a) SMA/PMMA(25/75) at 225°C • 0 o o o • ; ° % ° = " • 0 °*au 8 • o 6 B : / / . ° °a slope = 2 / . / * * / O n Preshear rate (s"1) " / / ° JP / o f t • 0.03 o 0.07 / o g D X • 0-1 * 0.3 ° °x w 0.7 O a o 1.0 1 1 1 1 1 1 1 1 1 1 1 1 ..1- 1_1.1.1.1 1 1 1 L_ 10-2 10"1 10° 10' 102 103 a>, rad/s 105 F 1Q2 I . . ' . . ' i i i i 11 , I 1Cr2 10-' 10° 101 102 10 co, rad/s Figure 6-7 Linear viscoelastic data (a: G ' ; b: G") for S M A 3 2 / P M M A 25/75 at 225°C with increasing preshear rate. CHAPTER 6 - NON-LINEAR RHEOLOGICAL RESPONSE OF PHASE SEPARATED SMA/PMMA BLENDS T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 111 The respective G ' and G" data with step-down preshear are depicted in Figure 6-8. The preshear rate and strains are also shown in Figure 6-8. Whereas the strains in this case differ from those of step-up preshear in Figure 6-7, the results are consistent in their nonmonotonicity in the same preshear rate ranges. co I * Q. b 10 3 J • ' 1 1 1 1 — 1 ' 1 1 • 11 1—I—I • I ' ' 11 1—1—1 • I 1111 1—= » A o * (b) SMA/PMMA(25/75) at 225°C • 0 o • • o • • • » v — V O © • w A . * O • V A O _ • ' A • O y A • y a O o A • V <• O ^ A : o o w * . • A O £ * 0 o e A / ? s A / slope = 1 A -o / • A / - i 10-' 10° 10' 10 2 co, rad /s Figure 6-8 Linear viscoelastic data (a: G ' ; b: G") for S M A 3 2 / P M M A 25/75 at 225°C with decreasing preshear rate. C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 112 As can be seen in Figures 6-7b and 6-8b, the terminal slope G'-co1 has been virtually reached at the various experiments, and thus the zero shear viscosity was extracted with reasonable accuracy. The dependence of the determined r\0 on the preshear strain is depicted in Figure 6-9. The rheogram exhibits a remarkable hysteresis taking into account the decay with time in material functions as discussed in detail below, although both the increasing and decreasing shear rate r\0 curves go through a minimum at about 0.3s"1. Referring to Figure 6-9, the fact that the viscosities at the highest shear rate or strain with increasing and subsequently decreasing shear are not identical, reflects the thermodynamic changes in the phase separating blend [relating to the spinodal decomposition and the associated morphological changes; see also Chopra et al. (1998), Bates and Wiltzius (1989)], which are distinguished from the fully immiscible systems; it is noted that between increasing and decreasing shear at the highest rate, the sample was annealed for 1 hour, during which the material functions change, as also reported by Chopra et al, (1998). This effect is attributed to the dependence of the blend morphology on the preshear history [Minale et al. (1997)]. The tangent dr^ldy in the inset of Figure 6-9 can be thought of as representing the rate of change in effective mass (alternatively mass rate in Figure 6-9) per unit length of the blend inclusions along the flow direction. A positive slope suggests an enhancement in the mass per unit length of the dispersed domains due to their deformation (maybe elongation), whereas a decrease in the slope is consistent with a scenario of transition in domain shape, in analogy to domain break-up during flow reported for the low viscosity inelastic blends studied by Minale et al. (1997); in the C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 113 present situation SEM evidence of quenched samples (e.g., Figure 6-4) along with the high viscosity ratio and glass transition contrast of dispersed and matrix phases suggest that shear leads to domain deformation rather than break-up, although the latter mechanism was recently reported [Lacroix et al. (1998)] for blends of polypropylene and ethylene copolymers possessing large viscosity contrast. Corroborating the findings of Figure 6-9, we speculate that at low shear rates, the shape of the inclusions is more curved (almost spherical), whereas at higher shear rates the curvature of the inclusions decreases (see also Figure 6-4; eventually at very high shear rates it becomes fibrillar, as discussed by Chopra et al. 1998) 10' 10 2 10 3 Y Figure 6-9 Dependence of zero shear rate viscosity (obtained from Figures 6-7 and 6-8) on shear strain and shear rate in the inset. Lines are drawn to guide the eye. C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 114 The minimum shear rate required to alter the shape of inclusions should depend on the "hardness" of the dispersed domains (actually it relates to the blend's dynamic asymmetry), the polarity of the constituents and the interfacial energy between the matrix and the inclusions. The composition of the inclusions and matrix in the SMA32/PMMA 25/75 blend at 225°C is SMA32/PMMA 65/35 (4>=0.05) and SMA32/PMMA 23/77, respectively [Chopra et al. (1998)]; the inclusions possess a higher Tg than the matrix, and as such they are more rigid. For this dispersion of SMA32-rich domains in a viscoelastic PMMA-rich matrix, a critical shear rate is needed to overcome the inertia of the high molecular weight constituents, favorable polar interactions and interfacial tension in order to deform the shape of the inclusions appreciably. This critical shear rate may be associated with a kind of metastable condition in terms of the morphology. The latter point was justified with the thixotropic loops discussed below. 6.3.3 Thixotropic Loops Figure 6-10 depicts typical thixotropic loop (shear rate ramps) measurements performed for the SMA32/PMMA 25/75 blend of Figure 6-9, at different final shear rate values. As can be seen in Figure 6-10a, hysteresis was observed for both shear rate loops of 0.3 s"1 and 0.5 s"1; on the other hand, no hysteresis was detected at shear rates 0.1s"1 and Is"1 (Figure 6-10b), conforming to a nonmonotonic behavior, analogous to that of Figures 6-9 and 6-7 and 6-8. It is noted that these rate ramps were reproducible (within 10%) with the same sample with increasing or decreasing final shear rate, and further they were all carried out at the same rate of change of y, to exclude contributions of different shearing gradients to the morphological response. C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 115 Figure 6-10 Thixotropic loops performed with S M A 3 2 / P M M A 25/75 at 225°C and different shear rates (arrows indicate direction of shear). C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 116 It is likely that in the vicinity of yKOs" 1 (corresponding to the viscosity minimum in the inset of Figure 6-9) the morphology is undergoing a transition from a nearly spherical (at rest) to elongated [Chopra et al. (1998, 1999)]. If the shearing process is stopped then the inclusions will return slowly to their equilibrium morphology. During the course of the deformation of the inclusions, the phase separated blend undergoes a decrease in the work of adhesion because of the decrease in the curvature of the elongated droplet. It can be deduced from Equation (6-1) that the loss in work of adhesion is compensated by the increase in interfacial tension. The "hysteresis area" in Figure 6-10a represents virtually the loss in work of adhesion per unit volume and unit time of the phase separated blend under the application of flow, which is due to the inability of the inclusions to have the same rate of curvature change in both increasing and decreasing shear rate processes. 6.3.4 Transient Rheological Response The shear and normal stress transients were modeled to a first approximation using Doi Ohta theory described in section 2.3 (Equations 2-21 and 2-22). Figure 6-11 depicts a typical result for the transient N\ response (at y=0.1 s"1) for the phase separated SMA32/PMMA 50/50 blend, along with the corresponding responses of the matrix SMA32/PMMA 25/75 and the inclusions SMA32/PMMA 60/40, with steady state M , eo /4o > M . 25 /75 , as well as the predictions of Equation (2-22), which are represented by the solid line. The volume fraction of inclusions was calculated a using lever rule and is equal to 0.71. The ratio a/d was determined as follows: The interfacial tension a was calculated from the analytical expression of Ermoshkin and Semenov (1996): C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 117 a =a„ 1-71 12X 1 1 • + K^SMA ^ PMMA J (6-4) where a» is the interfacial tension at infinite molecular weight, assuming a value here of 0.0048 N/m based on literature data [van Krevelen (1990)], and % is the Flory-Huggins interaction parameter calculated by Chopra et al. (1999). From these considerations, a value of ct=0.00137 N/m was obtained with a corresponding aJd=6.85 Pa, based on the SEM of Figure 6-4. 1 0 2 50 100 150 200 t, S 250 300 350 400 Figure 6-11 First normal stress difference transient response at y =0.1 s-1 for S M A 3 2 / P M M A 50/50 (o) and its constituent phases from the phase diagram, S M A 3 2 / P M M A 25/75 (•) and S M A 3 2 / P M M A 60/40 (•), at 220°C in the phase separated region. The solid line represents the fit of Equation (6-7). C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 118 It is apparent that the model captures the transient N\ response reasonably well. This agreement of theory and experimental data suggests indirectly that the steady state morphology of the phase separated SMA32/PMMA under shear is indeed fibrillar, i.e., corresponding to domain stretching in the flow direction, as also confirmed by SEM photographs [Chopra et al. (1998)]. It is noted, however, that the Doi-Ohta (1991) theory is not expected to work for viscosity ratios above 4, as in such cases the droplet size is no longer determined by the kinetics of flow only [Vinckier et al. (1996), Kitade et al. (1997)]. In the present case the viscosity ratio is much higher and further the dynamic asymmetry and the on-going thermodynamics affect the droplet size clearly; nevertheless, as a first approximation we use this approach by considering dispersed and matrix phases which are at the final stages of phase separation, i.e., long times [Chopra et al. (1998)]. Therefore, these results can be viewed as having a qualitative value only, and a more rigorous analysis of this problem with the correct assumptions concerning the growth of the droplets is clearly needed. The dependence of the first normal stress difference on shear rate and composition for the blends was investigated using steady shear measurements on the cone-plate rheometer. The relevant equations are mentioned in Appendix III. The first normal stress difference, N\, data can be used to model the corresponding shear-induced phase diagrams [Wolf (1984), Nafaile et al. (1984), Chopra et al. (1999b)]. Owing to equipment limitations, the latter material function is not an easy quantity to measure at high shear rates for blends of high viscosity like SMA32/PMMA. Laun (1986) proposed the following empirical relationship between the first normal stress coefficient, (=N\/y2) and the components of the complex modulus, G*(y), C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 119 V i = 2 G ' CO 1 + 2 A 0.7 G " (6-5) Figure 6-12 shows the N\ dependence on blend composition, ^ S M A 3 2 at 210°C, which is much below the LCST for SMA32/PMMA blends. This means that all data on Figure 6-12a refers to single-phase SMA32/PMMA blends. CO 10 7 10 6 10 5 10" 1 10 3 10 2 (a) 2 1 0 ° C f f f o 0.1s"' v 1.0s"1 • o - 10s"' 100s"1 Open' Experimental Closed: Laun's rule •J •I 0.0 0.2 0.4 0.6 0.8 1.0 Figure 6-12(a) The variation in first normal stress difference with increasing quantity .of SMA32 in S M A 3 2 / P M M A blend, at 210°C lying in one phase region. Figure 6-12b shows theM dependence on ^ S M A 3 2 at 230°C, which lies much above the LCST into the phase separated region. The predictions of Equation (6-5) are in excellent agreement with the experimental findings in the single-phase region (Figure 6-12a) and in reasonably good agreement with the experimental findings in the two-phase C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 120 region (Figure 6-12b). This gives us a starting point to model phase behavior of high viscosity phase-separated blends at high shear rates, where it is difficult to measure Ni. Such modeling was done by Chopra et al. (1999), where the N\ was experimentally measured. CO CL 1 0 7 p 1 0 6 1 0 5 10" 1 0 3 -i 1 1 1 1 1 1—i 1 1 r- ~i—i—~T 1—i—i—i 1 r-(b) 2 3 0 ° C O p e n : Exper imen ta l C l o s e d : Laun 's rule O 1.0s" 1 V 10s" 1 O - 100s" 1 -j—i—i—i—i—i—i—i—i—i—i i i i i i i i _J I I L_ 0.0 0.2 0.4 0.6 <t>SMA32 0.8 1.0 Figure 6-12(b) The variation in first normal stress difference with increasing quantity of SMA32 in S M A 3 2 / P M M A blend, at 230°C lying in 2 phase region. 6.3.5 Dynamic Interfacial Tension The interfacial tension is expected to be different when considering a fibril-type of morphology under (steady) shear, compared to a quiescent sphere-like morphology. In the context of 'dynamic interfacial tension', this means that this quantity can change with shear rate, but at the limit of zero shear rate it will yield its static equilibrium value. This can be rationalized by considering the thermodynamic definition of the interfacial C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 121 tension, a = ' where f/ is the internal energy of the system and A the interfacial area. ipA\ It is conceivable that the energy (in general the Helmholtz free energy) can have a contribution of a storage term, as long as this contribution is small enough to consider the system under quasi-equilibrium and thus apply the thermodynamic tools [this is a typical approach to study flow-induced phase transitions; see for example Chopra et al. (1999)]. Under these conditions, it is conceivable that the interfacial energy will change with flow, thus with curvature, and its value at high rates would correspond to elongated morphologies (say fibrils), whereas at the limit of vanishingly small shear rates is relates to the equilibrium nearly spherical morphology (Figure 6-4). An analogous effect of frequency dependence of a thermodynamic quantity, namely heat capacity (which at the zero rate limit yielded its static equilibrium value) has been observed in the context of glass transition dynamics [Dixon and Nagel (1988)]. By further assuming that the surface tension of both matrix and inclusion phases are independent of temperature in the range investigated (220°C-230°C) [Wu (1982)], the -in-dependent ratio oJd in the phase separated blend can be roughly estimated by fitting Equation (2-21) to transient stress data at high shear rates and treating ct/d as a parameter; the latter were obtained from transient shear experiments with the sliding plate rheometer. Typical results for SMA32/PMMA 50/50 corresponding to various shear rates are presented in Figures 6-13, 6-14 and 6-15 along with the corresponding fits. The decent agreement between predictions and experimental data once again favors the steady state fibrillar morphology of the inclusions. The so obtained values of a/d are plotted against C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 122 the shear rate in Figure 6-16; it was found that the data could be well represented by the following equation, which can be also rationalized from simple ideas based on Taylor's droplet deformation expressions [Utracki (1990)]: a . b — = ay + d ilibrium (6-6) where a and b are adjustable parameters which depend on viscosity and composition of the blend. For SMA32/PMMA 50/50, the values of a and b are mentioned in Figure 6-16. Figure 6-13 Shear stress transient response to start-up in simple shear flow for S M A 3 2 / P M M A 50/50 at 225°C (o) and 230 °C (v) and 1 s 1 along with the corresponding fits from Equation (2-21). C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 123 80 SMA/PMMA(50/50) t , S Figure 6-14 Shear stress transient response to start-up in simple shear flow for S M A 3 2 / P M M A 50/50 at 225°C (o) and 230 °C (v) and 10 s-1 along with the corresponding fits from Equation (2-21). 40 r 0.10 0.15 0.20 0.25 t, S 0.30 0.35 0.40 Figure 6-15 Shear stress transient response to start-up in simple shear flow for S M A 3 2 / P M M A 50/50 at 225°C (O) and 230 °C (v) and 100 s 1 along with the corresponding fits from Equation (2-21). C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 124 In consistency with the previous discussion, in the limit of vanishingly small shear rate the equilibrium value of a/d«6.85 Pa is reached. This result suggests the need for further experiments with other systems as well, to establish the interplay of shear rate and dynamic a. It is further noted that the transient shear stress response at y^lOs"1 and 230°C (Figures 6-14 and 6-15) exhibits a minimum, which may be indicative of the point where the inclusions undergo shape transition; this is probably a severe deformation (but not break-up), as supported by the large dynamic asymmetry. Further, the enhanced deviation between theoretical prediction and experimental data beyond the minimum may also reflect the fact that the theory considers a domain break-up mechanism, as already explained, in contrast to the experimental evidence. Figure 6-16 Dependence of the ratio <x/don the shear rate in die non-linear region. C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 125 Finally, it is noted that the steady state shear stresses of Figures 6-13, 6-14, 6-15 are much higher than the contribution of the components from linear additivity [Chopra et al. (1999)], suggesting that the stress response is dominated by the interfacial deformation, and thus justifying the use of Equations (2-21) and (2-22) which account for excess stresses. 6.3.6 Conclusion The nonlinear viscoelastic properties of SMA32/PMMA were found to be very sensitive to their phase behavior. The activation energy from the Arrhenius shift factors in the phase-separated region is higher in the linear regime because of the enhanced work of adhesion relating to the more stable morphologies. The effects of shear history on the morphological and rheological properties, the nonmonotonic dependence of the linear moduli on preshear rates and the hysteresis observed in the thixotropic tests corroborate the picture of shear-induced fibrillar growth of the SMA-rich inclusions in the phase separated regime. The stress evolution of phase separated blends was successfully described by a model combining the approach of Doi and Ohta with the affine deformation theory for single droplet behavior, supporting their steady state cylindrical morphology. These results indicate that the main rheological features of complex partially miscible polymer blends can be understood in terms of their thermodynamics and morphology using the relatively simple models of emulsions. C H A P T E R 6 - N O N - L I N E A R R H E O L O G I C A L R E S P O N S E O F P H A S E S E P A R A T E D S M A / P M M A B L E N D S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 126 7 Conclusions he application of a variety of thermal and rheological techniques allowed the thorough study of various PDMS/PEMS and SMA/PMMA blend systems. The phase behavior of these systems is very sensitive to the molecular structure of the individual components, in particular to molecular weight and presence of comonomers containing functional groups. Partial miscibility in the case of PDMS/PEMS arises from the similarity of chemical structure (enthalpy driven), whereas in the case of SMA/PMMA, is due to the presence of specific interactions between blend components. This work demonstrated the sensitivity of linear and non-linear rheology to the thermodynamic pretransitional forces in model dynamically symmetric and asymmetric blends. The viscoelastic properties of partially miscible blends can be modeled using simple models of Palierne (1990) and the approach of Vinckier et al. (1997a), which describe the rheology of emulsions of two incompressible viscoelastic blends, coupled with the precise knowledge of the phase behavior of the blends. Conclusions for PDMS/PEMS blend system PDMS/PEMS blends containing PDMS withM w in-between 5-10 kg/mol showed-UCST behavior with critical temperatures between 25°C-150°C. According to theoretical predictions based on Flory-Huggins theory, critical temperature increases as the molecular weight of PDMS increases. The phase separation temperatures were determined rheologically by noting the change in slope in a stress versus temperature in a steady state shear experiment. The phase separation mechanism, which happens via C H A P T E R 7 - C O N C L U S I O N S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 127 spinodal decomposition in PDMS/PEMS blends, was observed by optical microscopy. In the phase separated region, droplet/matrix morphology was observed. Estimates of interfacial tension for the phase separated blends were obtained by using the Palierne emulsion model and the Helfand-Tagami theory. Interfacial tension and the radius of the dispersed droplets decreased upon reduction of the Mw of PDMS implying improved compatibility. Conclusion for SMA/PMMA blend system The phase behavior of SMA/PMMA binary blends as a function of M A content in the SMA copolymer was investigated. Miscibility in these blends was established by the presence of a single glass transition temperature (at temperatures below 200°C) exhibiting a breadth below 30°C. It was demonstrated that as the miscibility of the blend increases, the breadth of glass transition decreases. Rheological detection of the phase separation temperatures (binodal) was based on the increase in elastic modulus and presence of a peak in tand during dynamic temperature sweeps, appearance of dual peaks on the onset of phase separation in a Cole-Cole plot and failure of time-temperature superposition in dynamic frequency sweeps in the phase separated region. The magnitude of tand peak could provide a qualitative measure for comparing the miscibility of various blends, with higher miscibility being characterized by higher tand peaks. Experimental phase diagrams of SMA/PMMA using the above criteria exhibited a lower critical solution temperature. These phase diagrams were successfully described using the lattice treatment based on the Flory-Huggins theory and assuming only temperature dependence of the interaction parameter. SMA/PMMA blends display higher C H A P T E R 7 - C O N C L U S I O N S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 128 miscibility as M A content increases from 8% to 14% due to the increased interaction between the M A group and PMMA. However, increase in M A content beyond a certain point results in a reduction in miscibility, apparently because inter-molecular interactions become more dominant. In summary, the above methodological approach allows the study of the phase behavior of a variety of polymer blends, including industrially important systems which typically are characterized by high glass transition, high molecular weight, and varying degree of dynamic asymmetry. Conclusion for SMA32/PMMA blend system Linear and non-linear viscoelastic measurements were carried out with a LCST blend of high T$ and molecular weight, namely poly(styrene-co-maleic anhydride) and poly(methyl methacrylate), SMA32/PMMA, over a wide range of shear rates in the phase separating region, using parallel plate, sliding plate and capillary rheometry. The main findings of this investigation are summarized as follows: The values of wall shear stress obtained from shear experiments on a sliding plate rheometer were lower than those obtained from previous capillary experiments; the difference was attributed to the effect of pressure on viscosity. LAOS experiments conducted on the phase separated blends revealed the presence of an additional relaxation mechanism (a second peak) which was not observed in the homogeneous region and may thus be another sensitive indicator of phase separation. C H A P T E R 7 - C O N C L U S I O N S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 129 The nonlinear viscoelastic properties are sensitive to the phase behavior much like their linear analogues; the failure of time temperature superposition represented the signature of the onset of phase separation. The activation energy from the Arrhenius shift factors in the phase separated region was lower in the non-linear regime apparently due to the presence of more deformed domains in the nonlinear regime, as evidenced by SEM. This was supported by comparison of LAOS, small amplitude oscillatory shear and capillary data. The nonmonotonic dependence of the linear viscoelastic properties on the blend's preshear rates and the hysteresis observed in the thixotropic loops corroborate the postulate of shear-induced fibrillar growth of the SMA32-rich inclusions in the phase separated regime. The time evolution of shear and normal stresses in the phase separated blend during transient simple shear flow at shear rates as high as 100 s"1 was modeled according to the approach of Vinckier et al. (1997b), combining the theory of Doi and Ohta with the affine deformation theory for single droplet behavior. The good agreement at low shear rates supports their steady state cylindrical morphology. The key message from the above results is that the main rheological features of complex partially miscible polymer blends can be understood to a first approximation in terms of their thermodynamics using the relatively simple models of emulsions and immiscible blends, provided that their phase state is known with accuracy C H A P T E R 7 - C O N C L U S I O N S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 130 7.1 Contributions to Knowledge A thorough investigation of linear and non-linear rheological properties of UCST and LCST partially miscible blends in both the phase separated and single-phase regions has been performed. The program resulted in several significant contributions to knowledge. They are listed as follows: • For the first time PDMS/PEMS blends made with high molecular weight PEMS were subject to a systematic rheological and optical study to find out the effect of molecular weight of PDMS on its phase behavior and morphology at different temperatures. • It was shown that the activation energy of monodisperse PDMS and PEMS is independent of molecular weight. • In addition to well-established rheological methods for detecting phase separation in partially miscible blends, two new methods were found viz. peak in tan8 in a linear viscoelastic temperature sweep and detection of additional relaxation (double peaks) in phase separated blends by LAOS. • The effect of level of shear on the morphology of phase separated partially miscible blend (SMA32/PMMA) was evaluated by determining activation energy from the Arrhenius shift factors. Furthermore, the occurrence of a shear-induced fibrillar growth of droplets in the phase-separated region was confirmed by performing preshear frequency sweeps and observing a hysteresis in thixotropic loops in the linear viscoelastic regime. The finding was confirmed with SEM. • Transient stresses (M and o~) in the phase-separated blend were successfully modeled using the approach of Vinckier et al. (1997b) in conjunction with levers rule. The C H A P T E R 7 - C O N C L U S I O N S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 131 model was used to understand the effect on dynamic interfacial properties of partially miscible phase separated blend for the first time. • The non-monotonic effect of M A content on the phase diagram of SMA/PMMA blends and on the enhanced concentration fluctuations in the pretransitional regime was explained in terms of its thermodynamics. 7.2 Recommendations Based on the experience gained during this study, the following recommendations for future work can be made. • Investigate the effect of shear on phase separation dynamics in partially miscible blends using the optical shearing stage. This can lead to a better understanding of shear induced changes in phase behavior of SMA/PMMA blends reported in literature [Aelmans et al. (1999), Chopras al. (1998, 1999)]. • Improve the shear-induced phase behavior prediction approach of Chopra et al. (1999) based on Flory-Huggins model by incorporating other measures of anisotropy like birefringence. This would give way to the problems associated with the determination of N\ at high shear rates. Also it would help in confirming that mixing and demixing in the same blend are primarily of entropic origin. • Improve understanding of the instabilities, droplet break-up and coalescence associated with the thermo-rheologically, complex behavior of various phase-separated systems using optical shearing equipment. 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R E F E R E N C E S T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 142 APPENDIX I - Sliding Plate Rheometer For this work, an Interlaken sliding plate rheometer with a flush-mounted shear stress transducer was used. The basic features of a sliding plate rheometer are shown in Figure 1-1 [Giacomin et al. (1989)]. An end plate is acted on by the shear stress generated by the fluid and transmits the resulting moment to the cantilever beam. To avoid melt penetration into the gap around the end plate, the deflection of the latter must be limited to very small levels. That is why a capacitance system was used, where a capacitor is formed by the probe acting as one of the plates, and the beam as the second plate. There are many advantages associated with the direct measurement of the shear stress: • Uncontrolled flow at the edges of the sample does not affect the determination of the shear stress, allowing tests with large and rapid deformation to be carried out; Figure 1-1. Schematic diagram of the shear stress transducer. Degradation occurring as a result of contact between the exposed edges of the sample and the environment does not affect the measurement; A P P E N D I X I T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 143 • The exact size and shape of the sample need not to be known, and this greatly simplifies sample loading; • Tests can be carried out with only a few grams of sample; Bearing friction has no effect on the measured shear stress, as long as it does not introduce mechanical noise. Besides steady shear measurements a sliding plate rheometer can be used to generate large amplitude oscillatory shear [LAOS] flow. The equations in previous section on simple shear flow are fully applicable to the analysis of the data obtained by means of this instrument. The equations corresponding to LAOS data analysis are discussed in detail in section 6.3.1. A P P E N D I X I T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 144 APPENDIX II - Parallel Plate Rheometer Measurements of rheological properties at low shear rates and deformations are usually carried out in rotational rheometers such as the cone-and-plate or parallel-plate rheometers (Figure II-1). The two plates are mounted on a common axis of symmetry, and the sample is inserted in the space between them. The upper plate is rotated at a specified angular velocity co(t) and as a result the sample is subjected to shear. The motion of the upper plate is programmed, and the resulting torque, M, is measured (so-called constant strain rheometers). Another mode of operation is fixing the torque and measuring the displacement (constant-stress rheometers). Reproducibility of such devices lies within ±2%. The most widely used experiments to determine the linear viscoelastic properties of polymers are small amplitude oscillatory shear tests. In this experiment, a sample of P r e s s u r e " t r a n s d u c e r Figure I I - l Parallel plate rheometer A P P E N D I X II T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 145 material is subjected to a simple shear ring deformation such that the shear strain is a function of time given by: y(0 = y 0 sin(firf) (II-l) where y0 is the strain amplitude and co is the frequency. The stress is then measured as a function of time. It can be shown that the shear stress is sinusoidal in time and independent of strain (small linear viscoelastic limit): a(0 = a 0 sin( co t + 8) (II-2) where a 0 is the stress amplitude and 8 is a phase shift, or the mechanical loss angle. Using a trigonometric identity, one can rewrite Equation (II-2) in the following form: <*(0 = Y0 [G'O) sin(©/) + G'(co) cos(co t)] (II-3) where G'(co) is the storage modulus and G"(co) is the loss modulus. These two quantities can be calculated from the amplitude ratio, Gd =<J0/y0 , and the phase shift, 8, as follows: G' = Gd cos(8) (II-4) G" = Gd sin(8) (II-5) This allows defining a complex modulus, G*(co), as follows: G*(co) = G'(«)) + iG ' ( (B) (II-6) Alternatively, the stress can be expressed in terms of two material functions, rf and rf \ having units of viscosity as follows: °(0 = Y0[rl'(©)cos(coO + ri'(co)sin(coO] (H-7) thus defining the complex viscosity: r]*(co) = ri'(co) - /^ "(co) (II-8) A P P E N D I X II T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 146 APPENDIX III - Cone and Plate Rheometer A small angle cone is mounted on a common axis with a circular disk as shown in Figure III-1. The sample is inserted between them and one element is rotated and the other is held stationary. Figure I I I - l Cone and plate rheometer The shear rate in cone-and-plate rheometer experiments is given by the following expression [Dealy (1982)]: where co is a rotational speed, and Go is the cone angle. For a body of fluid undergoing laminar flow, the shear rate it is experiencing is the ratio of the velocity difference across the fluid to the distance over which the shearing occurs. At point, PI i.e. at the rim of a cone, the shear rate experienced by the sample beneath that point is the ratio of the speed of that point to the sample thickness at that point. Since, the cone has a shallow angled (typically less than 4°), this ratio remains constant at any radius out from the tip. For example, a point P2 on the underside of the cone at, say, half the radius, will A P P E N D I X III T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 147 be traveling at half the speed but the sample gap will also be halved. A uniform shear rate is therefore generated across the entire sample for any given rotational speed. The torque required to rotate the cone (or to hold the plate stationary) is given by r M = J x ^ 2%r2dr = r 3 x e + (III-2) o where r^is the shear stress, r is a cone radius. Thus, the value of the viscosity at one shear rate can be computed from the results of a single experiment: *e* 3M9 0 Tl = = 5 (III-3) y 27t r co v ' The stress relaxation modulus, G(t), from step strain experiments can be calculated as G ( 0 = * e * O A (III-4) where y is the shear strain. The storage modulus, G\ and the loss modulus, G", can be determined from small amplitude oscillating experiments as follows: , 36 0-^oCosS G = ° (III-5) 39 n M f t sin 5 (III-6) where Mo is a torque oscillating amplitude, fa is angular amplitude, and 8 is the loss angle. The first normal stress difference, N\, is given by the following equation, IF N i = (m-7) A P P E N D I X III T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 148 where F is the normal force. Equation (III-7) neglects the contribution to the normal force due to centripetal acceleration. However, for melts within the range of usable shear rates in a cone-plate rheometer, this contribution is negligible. A P P E N D I X III T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 149 APPENDIX IV - Capillary Rheometer The most widely used type of melt rheometer is the capillary rheometer. This device (Figure IV-1) consists of a melt reservoir, or barrel, for melting the polymer and a plunger or piston that causes the melt to flow through the capillary die of known diameter, D, and length, L. Electric heaters > < _ JL Un Constant force or constant rate 10 Piston E i -Melt reservoir -Tef lon O-ring Thermocouple —Capil lary die Figure TV-1 Capillary rheometer The quantities normally measured are the flow rate, Q, (related to the piston speed) and the driving pressure, AP, (related to force on the piston that is measured by means of a load cell). Reproducibility of capillary rheometers is ±5%. Capillary rheometers are used primarily to determine the viscosity in the shear rate range of 5 to 1,000 s"1. To calculate the viscosity, one must know the wall shear stress and the wall shear rate. For steady-state, fully-developed flow of an incompressible Newtonian fluid, the wall shear stress, <jw, can be calculated as: A P P E N D I X IV T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 150 APD - . - ^ (iv-i) The magnitude of the wall shear rate, yw, for a Newtonian fluid can be calculated as: 320 KD3 t w = 3 § - OV-2) For the case of a non-Newtonian fluid, this quantity is called the apparent wall shear rate, f A, that is the rate that a Newtonian fluid would have at the same volumetric flow rate Q: 32Q 7tD3 Capillary flow of a Newtonian fluid is a controlable flow which means that the flow kinematics does not depend on the nature of the fluid. Capillary flow of molten polymers, however, is only a partially controllable flow. This means that the velocity distribution in this flow is governed not only by the boundary conditions but also depends on the nature of the fluid. To account for this, at least two corrections should be applied to the experimental data. First, the velocity profile in the flow of a polymeric fluid is nonparabolic, and one must correct the wall shear rate, yw, defined by Equation (IV-2). This correction, generally known as the Rabinowitch correction, can be calculated as [Dealy and Wissbrun (1990)]. b = ^ M A ( I V -4) It is noted that the correction factor b is a local quantity depending on yA. It can be shown that the true wall shear rate then can be obtained by use of the following equation [Dealy and Wissbrun (1990)]: '3 + ^ Yv (IV-5) V t J A P P E N D I X IV T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 151 For a power-law fluid, the shear stress is given by a = Ky" (IV-6) where a is the shear stress, y is the shear rate, K is the consistency index, and n is the power law exponent. It can be shown that the wall shear rate for a power law fluid can be expressed as: ^3 + l/«^ ft A OV-7) v "* J 4 Thus, it can be seen from Equations (IV-5) and (IV-7) that the Rabinowitch correction is equal to \ln for a power law fluid and 1 for a Newtonian fluid. It mainly represents deviation from Newtonian behavior. Second, the pressure drop must be corrected for the additional pressure required for the melt to pass through the contraction between the barrel and the capillary. There is a significant pressure drop near the entrance of the die, APenl. There may also be residual pressure at the die exit, called the exit pressure, A P e x , but it is quite small compared to APent- The total pressure correction for exit and entrance regions is called the end pressure, APead, that is, A P ^ (IV-8) The true wall shear stress is then obtained as: (AP-AP ,) ( I V " 9 ) The pressure correction, APend, or the Bagley end correction can be determined by use of a scheme proposed by Bagley (1957). He suggested to measure the driving pressure, Pd, at various values of the flow rate, Q, using a variety of capillaries having different A P P E N D I X IV T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 152 lengths. For each value of the apparent wall shear rate [Equation (IV-3)] he then plotted driving pressure versus LID and drew a straight line through the points as shown in Figure IV-2. Figure IV-2. Bagley plot for determining the end correction. Extrapolating the lines to the Pj=0 axis, he then obtained an end correction, e. Thus, the true wall shear stress can then be calculated as follows: °W = % , (IV-10) 4(L/D + e) An alternative way to directly measure APend is to use orifice capillaries with L/D=0. In this case, the true wall shear stress can be found by applying Equation (IV-9). More details on capillary rheometry including a discussion of viscous heating, pressure effects, wall slip, and polymer degradation can be found in Dealy (1982) and Dealy and Wissbrun (1990). A P P E N D I X IV T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 153 Notation a fraction of mean scattering intensity, dimensionless a,b adjustable parameters in Equation (6-6), dimensionless a? shift factor, dimensionless A empirical parameter, dimensionless b Rabinowitsch correction, dimensionless bi statistical segment length of component /', A B empirical parameter, K Ci, C2 empirical WLF parameters, dimensionless d diameter of the inclusion, m D capillary diameter, m e Bagley end correction, dimensionless f* light scattering instrumental factor, dimensionless F Normal force, N g field autocorrelation function, dimensionless G shear modulus, Pa G' storage modulus, Pa G" loss modulus, Pa G* complex modulus, Pa G°N plateau modulus, Pa A G m molar change in Gibbs free energy of mixing, J / mol A G M change in Gibbs free energy per segment upon mixing, J h gap between the plates, m AHM enthalpy of mixing, J AH Heat change per unit time in Figure 5-2a, mW I Intensity of light, arbitrary units N O T A T I O N T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 154 k Boltzmann' s constant (1.38x10"23 J / K) K ratio of glass transition temperatures of two polymers in the blend, dimensionless L capillary or slit length, m M Torque, N . m Mc critical molecular weight for entanglement, kg / mol M e average molecular weight between entanglements, kg / mol M„ number average molecular weight, kg / mol M0 molecular weight of monomer unit of PEMS, kg / mol Mw weight-average molecular weight, kg / mol n power law exponent, dimensionless n harmonic in LAOS data analysis i.e. in Equation (6-2) and (6-3), dimensionless ttCeii total number of cells in the lattice, dimensionless N number of data points in LAOS experiment, dimensionless N\ degree of polymerization of polymer (i = 1,2), dimensionless No Avogadro's number (6.022 x 1023 mol"1) N\ first normal stress difference, Pa P absolute pressure, Pa Pa ambient pressure, Pa Pd driving pressure, Pa APend Bagley correction, Pa APex exit pressure drop, Pa APent entrance pressure drop, Pa q magnitude of scattering wave vector, nm"1 q empirical parameter used in Equation (5-1), K Q volumetric flow rate, m3 / s N O T A T I O N T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 155 R gas constant (8.31451 J / mol. K) R radius of the droplet, m r radius of cone, m r, number of segments per chain, dimensionless s entropy of system, J / K ASc combinatorial entropy of mixing, J / K A 5 N C non-combinatorial entropy of mixing, J / K ASM entropy of mixing, J / K t time, s T absolute temperature, K Critical critical temperature of the blend, K Td decomposition temperature, °C Tg glass transition temperature, °C Tref reference temperature, °C 7w breadth of glass transition, °C u melt velocity, m / s U internal energy, J V segment volume, m3 Wa work of adhesion, J Ax plate displacement, m z discrete frequency component index, dimensionless Greek Letters a interfacial tension, N / m ao pressure coefficient of viscosity, Pa"1 ct°o interfacial tension at infinite molecular weight, N / m N O T A T I O N T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y MISCIBLE P O L Y M E R B L E N D S 156 X interaction parameter, dimensionless Xc value of interaction parameter at the critical temperature, dimensionless maximum value of interaction parameter, dimensionless 6 mechanical loss angle, degrees 8 parameter for extent of concentration fluctuations, dimensionless 4>c critical volume fraction of the blend, dimensionless <j>dr volume fraction of droplets, dimensionless <t>i volume fraction of the blend components (i = 1, 2) , dimensionless <|>im mole fraction of the blend components (i = 1, 2) , dimensionless Y(0 shear strain, dimensionless yv rate of deformation tensor, s"1 Y shear rate, s"1 y A apparent shear rate, s"1 Y w wall shear rate, s"1 Yo strain amplitude in oscillatory shear, dimensionless viscosity, Pa • s "no zero-shear viscosity, Pa • s T,(0) viscosity at ambient pressure, Pa • s i dynamic viscosity, Pa • s out-of-phase component of complex viscosity. Pa • s T l A apparent viscosity, Pa • s X characteristic time of material, s M- Chemical potential, J / mol first normal stress coefficient, Pa . s2 P density, kg / m3 a shear stress obtained from sliding plate rheometer, Pa a d discrete shear stress, Pa N O T A T I O N T H E R M O D Y N A M I C S A N D R H E O L O G Y O F P A R T I A L L Y M I S C I B L E P O L Y M E R B L E N D S 157 a n shear stress corresponding to harmonic, Pa rjw wall shear stress, Pa o 0 stress amplitude in oscillatory shear, Pa Tjj stress tensor components (/' = x, y, z; j = x, y, z), Pa co frequency, rad / s N O T A T I O N 

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