©2007 Rushikesh. A. Matkar All Rights Reserved

Home , Eutectic system

RUSHIKESH. A. MATKAR

PHASE DIAGRAMS AND KINETICS OF SOLID-LIQUID PHASE TRANSITIONS IN CRYSTALLINE POLYMER BLENDS

A Dissertation

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy

Rushikesh A. Matkar

December, 2007

PHASE DIAGRAMS AND KINETICS OF SOLID-LIQUID PHASE TRANSITIONS IN CRYSTALLINE POLYMER BLENDS

Rushikesh A Matkar

Dissertation

Approved: Accepted:

______Advisor Department Chair Dr. Thein Kyu Dr. Sadhan C. Jana

______Committee Member Dean of the College Dr. Chang D. Han Dr. Stephen Z. Cheng

______Committee Member Dean of the Graduate School Dr. Avraam I. Isayev Dr. George R. Newkome

______Committee Member Date Dr. Gustavo A. Carri

______Committee Member Dr. Dmitry Golovaty

______Committee Member Dr. Subramaniya I. Hariharan

ii ABSTRACT

A free energy functional has been formulated based on an order parameter approach to describe the competition between liquid-liquid phase separation and solid-liquid phase separation. In the free energy description, the assumption of complete solvent rejection from the crystalline phase that is inherent in the Flory diluent theory was removed as solvent has been found to reside in the crystalline phase in the form of intercalates. Using this approach, we have calculated various phase diagrams in binary blends of crystalline and amorphous polymers that show upper or lower critical solution temperature. Also, the discrepancy in the χ values obtained from diﬀerent experimental methods reported in the literature for the polymer blend of poly(vinylideneﬂuoride) and poly(methylmethacrylate) has been discussed in the context of the present model. Experimental phase diagram for the polymer blend of poly(caprolactone) and polystyrene has also been calculated. Of particular importance is that the crystalline phase concentration as a function of temperature has been calculated using free energy minimization methods instead of assuming it to be pure. In the limit of complete immiscibility of the solvent in the crystalline phase, the Flory diluent theory is recovered. The model is extended to binary crystalline

iii blends and the formation of eutectic, peritectic and azeotrope phase diagrams has been explained on the basis of departure from ideal solid solution behavior. Ex- perimental eutectic phase diagram from literature of a binary blend of crystalline polymer poly(caprolactone) and trioxane were recalculated using the aforementioned approach.

Furthermore, simulations on the spatio temporal dynamics of crystallization in blends of crystalline and amorphous polymers were carried out using the Ginzburg-

Landau approach. These simulations have provided insight into the distribution of the amorphous polymer in the blends during the crystallization process. The simulated results are in close accordance with the experimentally observed concentration proﬁles of atactic polypropylene during the crystallization of isotactic polypropylene in a blend of these polymers. Finally described are the unique thermodynamics and kinetics that occur in thermoplastic elastomer blends of polypropylene and synthetic rubber, leading to the unique biphasic crystalline morphology imparting these blends with their characteristic high toughness and high impact strength. Phase diagrams in such blends exhibit a combined upper and lower critical solution temperature.

These phase diagrams have been calculated based on the present model developed, and simulated results explain the structural development in these blends.

iv ACKNOWLEDGEMENTS

This dissertation would not have been possible without the love and guidance of my parents, Hema and Ashok Matkar. Their faith and conﬁdence in my abilities has helped me achieve my goals. I am indebted to Dr. Thein Kyu who has been a mentor, teacher and a guardian during the course of my studies. He has been a source of inspiration both professionally and personally. I am thankful to the department of polymer engineering for admitting me to the graduate program and granting a research assistantship. I would like to thank Dr. Chang D. Han, Dr. Avraam I. Isayev,

Dr. Gustavo A. Carri, Dr. Dmitry Golovaty and Dr. Subramaniya I. Hariharan for the constructive criticisms, helpful suggestions during the undertaking of this research and for serving on my committee. As a graduate student, I have had several fruitful discussions with my colleagues (Dr. Pratush Dayal, Mr. Pankaj Rathi, Mr.

Atanas Gagov) and outstanding friends (Dr. Ranjan Grover, Dr. Subal Sahni, Mr.

Christopher Rottmayer) who have contributed signiﬁcantly towards the contents of this dissertation. I would like to thank my older brother, Rutuparna Matkar, who has always been an inspiration and my wife, Emily, for providing constant unconditional love.

v TABLE OF CONTENTS

Page

LIST OF FIGURES ...... xi

CHAPTER

I. INTRODUCTION ...... 1

II. LITERATURE REVIEW ...... 6

2.1 Basic Thermodynamic Relations ...... 6

2.1.1 Closed Systems ...... 6

2.1.2 Open Systems ...... 7

2.2 Classical Thermodynamics of Crystallization ...... 10

2.2.1 Equilibrium Theory of Crystallization ...... 10

2.2.2 Gibbs-Thompson Equation ...... 13

2.2.3 Week-Hoﬀman Plots ...... 14

2.3 Flory-Huggin Theory ...... 14

2.3.1 Entropy of Mixing of Simple Molecules ...... 15

2.3.2 Entropy of Mixing of Polymers ...... 15

2.3.3 Enthalpy of Mixing of Polymers ...... 16

vi 2.4 Determination of Phase Equilibria ...... 19

2.4.1 Gibbs Criteria ...... 19

2.4.2 Liquid-Liquid Mixtures ...... 21

2.4.2.1 Spinodal Region and Critical Points ...... 22

2.4.2.2 Binodal Region: Michelson’s Method ...... 24

2.4.3 Solid-Liquid Mixtures ...... 27

2.4.3.1 Prigogine’s Model ...... 27

2.4.3.2 Flory Diluent Theory ...... 29

2.4.3.3 Estimation of χ Parameter ...... 31

2.5 Thermodynamics of Crystallization Based on Order Parameter Approach ...... 33 2.5.1 Crystallization in Small Molecule Systems ...... 34

2.5.2 Relationships Between Model Parameters and Material Properties for Small Molecule Systems ...... 36 2.6 Phase Equilibria in Binary Crystalline Blends ...... 38

2.6.1 Ideal Crystalline Blends ...... 39

2.6.2 Eutectic Systems ...... 41

2.7 Experimental Phase Diagrams ...... 45

2.7.1 Poly(vinylideneﬂuoride)/Poly(methylmethacrylate) Blends . 46

2.7.2 Poly(caprolactone)/Polystyrene Blends ...... 48

2.7.3 Blends of Isotactic Polypropylene and Ethylene Propy- lene Diene Monomer ...... 48 2.7.4 Blends of Syndiotactic Polypropylene and Ethylene Propylene Diene Monomer ...... 52

vii 2.7.5 Blends of Polyethylenes of Diﬀerent Molecular Weights . . . 54

2.7.6 Poly(caprolactone)/Trioxane Blends ...... 54

2.8 Dynamics of Crystallization of Homopolymer ...... 56

2.8.1 Avraami Model ...... 56

2.8.2 TDGL Model A ...... 62

2.8.3 Role of Latent Heat ...... 64

2.9 Dynamics of Liquid-Liquid Demixing in Binary Mixtures . . . . . 67

2.9.1 Nucleation and Growth (NG) ...... 68

2.9.2 Spinodal Decomposition (SD) ...... 70

2.10 Dynamics of Crystallization in Polymer Blends ...... 75

III. BLENDS OF CRYSTALLINE AND AMORPHOUS POLYMERS . . 79

3.1 Introduction ...... 79

3.2 Free Energy Landscape ...... 81

3.3 Completely Miscible Systems ...... 86

3.4 Partially Miscible Systems: ...... 94

3.5 Conclusions ...... 97

IV. BINARY CRYSTALLINE BLENDS ...... 100

4.1 Introduction ...... 100

4.2 Phase Field Free Energy of Crystallization of a Homopolymer . . . 102

4.3 Extension of the Phase Field Model of Crystallization to Poly- mer Blends ...... 105 4.4 Construction of Phase Diagrams via Free Energy Minimization . . 107

viii 4.5 Results and Discussion ...... 111

4.6 Comparison with Experiment ...... 115

4.7 Conclusions ...... 120

V. DYNAMICS OF CRYSTALLIZATION IN CRYSTALLINE AND AMORPHOUS BLENDS ...... 123 5.1 Introduction ...... 123

5.2 Review: Thermodynamic Model ...... 126

5.3 TDGL Model C Equations ...... 129

5.4 Results and Discussion ...... 131

5.5 Conclusion ...... 137

VI. PHASE DIAGRAMS AND MORPHOLOGY EVOLUTION IN PP/EPDM BLENDS ...... 139 6.1 Introduction ...... 139

6.2 FH Theory for UCST and LCST Systems ...... 142

6.3 Prediction of Phase Diagram Topologies ...... 143

6.4 Comparison with Experimental Phase Diagrams of PP/EPDM Blends ...... 147 6.5 Dynamics of Crystal Growth in a Phase Separating System: iPP/EPDM Blends ...... 152 6.6 Dynamics of Crystal Growth in a Phase Separating System: sPP/EPDM Blends ...... 157 6.7 Conclusions ...... 162

VII. CONCLUSIONS AND RECOMMENDATIONS ...... 168

7.1 Conclusions ...... 168

ix 7.2 Recommendations ...... 171

BIBLIOGRAPHY ...... 175

APPENDICES ...... 185

APPENDIX A. COMMON TANGENT METHOD ...... 186

APPENDIX B. EMPIRICAL RELATIONS OF χ ...... 190

APPENDIX C. CONVERSION OF MATERIAL TO MODEL PA- RAMETERS ...... 194

x LIST OF FIGURES

Figure Page

2.1 Free energy of mixing of a hypothetical binary liquid mixture with A = 0, X1 = 1 and X2 = 1 with Tcrit = 300K has been plotted as a function of composition for various temperatures to illustrate the appearance of a double well structure below the critical temperature that leads to UCST type liquid-liquid phase separation. For temperatures above Tcrit (350K and 325K) the free energy is a single well and the mixture is stable. As temperature decreases below 300K a local maximum at φ = 0.5 is discerned that increases with decreasing temperature. The local maximum indicates that the mixture is unstable according to the Gibbs criterion and the equilibrium compositions of the phases can be found by the common tangent method. The dotted line plotted for the free energy of mixing at T = 225K is the common tangent line and the points at which it touches the free energy curve are the equilibrium compositions. 20 2.2 Phase diagram exhibiting an upper critical solution temperature has been calculated using the Flory Huggins theory of isotropic mixing to illustrate the binodal (dark black line) and the spinodal regions (gray line) using the parameters A = 0, X1 = 1 and X2 = 1 with Tcrit = 500K. The mixture exists as a single phase outside the binodal envelope and is termed stable. The equilibrium compositions that determine the binodal envelope are calculated using the common tangent approach. Additionally a spinodal envelope is calculated within the binodal envelope by setting the Gibbs inequality in Equation (2.30) to zero. The region between the binodal and spinodal envelope is termed metastable ...... 23

xi 2.3 Michelson’s method to ﬁnd the binodal points on the phase diagram using the tangent plane minimization approach [25]. For a binary system, this method uses a probe concentration φz and uses a slope matching algorithm to ﬁnd concentrations φs1 and φs2. If the probe concentration is within the binodal then it is never the lowest tangent line, D(φs1, φz) < 0 or D(φs2, φz) < 0. These concentrations of the liquid mixture are unstable. If both are greater than zero, then the mixture is stable. By mapping this transition from stable to unstable we can determine the equilibrium composition (φα and φβ) of the two phases...... 25 2.4 Illustration of the graphical procedure to equate the chemical potential of the solid and liquid phases to determine the phase diagram . 32

2.5 Local free energy of crystallization f(ψ) is represented by a Landau expansion in ψ. A fourth order polynomial with coeﬃcients representing a double well structure is chosen. The coeﬃcients are related to the heat of crystallization and the supercooling (∆T ) imposed on the system. ψ = 0 represents the melt and ψ = 1 represents the 0 0 crystal. At T < Tm, crystal is more stable than melt and at T = Tm free energy of both solid and melt are the same and equal to zero. 0 At T > Tm, the melt is at a lower minimum than the crystal. ζ marks the location of the local maximum and f(ζ) is analogous to the nucleation barrier to crystallization...... 37 2.6 The ideal solid solution phase diagram of a binary crystalline alloy of Copper (Cu) and Nickel (Ni) as established by McFadden et al. [38]. Zero on the abscissa represents pure Ni and one represents pure Cu. The melting temperatures of pure Cu is 1358K and pure Ni is 1728K. Above 1728K , the blend exists in the melt state at all concentrations. At temperatures lower than the melting temperature of Ni, the equilibrium concentration of the solidus and liquidus phases is calculated using the common tangent algorithm to the free energy described in Equation (2.51). The dotted line denotes the crystal-melt transition where the free energy of crystal and melt are equal. 42

xii 2.7 Thermodynamics of eutectic phase diagrams using the order parameter approach. The phase diagram has been established as a function of ∆T with concentration order parameter (ς). ς is deﬁned as φ − φ0 where φ0 is the initial concentration of the blend. Various coexistence regions (Solid-Liquid) and (Solid-Solid) were calculated using the free energy described by Elder [43, 44]. Two solid-liquid coexistence regions intersect to form the eutectic where the liquid is in equilibrium with two solid phases. Below the eutectic point lies a region of Solid-Solid immiscibility...... 43

2.8 Melting point depression of PVDF/PMMA blend as function of PMMA volume fraction as determined by Nishi and Wang [26]. Open Symbols denote experimental data for the observed melting point depression using Dynamic Scanning Calorimetry (DSC). Solid squares indicate the experimantally determined glass transition temperatures (Tg) which are quite invariant to composition. Single Tg is evidence of the miscible nature of the mixture at all compositions...... 47

2.9 Experimentally determined phase diagram of the PCL/PS blend by Tanaka and Nishi [53]. The solid symbols are the experimental data for the melting temperature and the open symbols are the phase separation temperatures obtained from cloud point measurements. Solid lines are ﬁts computed using the Flory diluent theory and the Flory-Huggins mixing theory...... 49

2.10 Experimental phase diagram of the iPP/EPDM blend determined by Chen and Kyu [54] using light scattering and DSC techniques, showing the intersection of LCST and the crystal-melt phase transition. The solid circle symbols denote the melting temperature of the blend at various compositions obtained from heating run on the DSC trace. The solid square symbols are the DSC crystallization peaks obtained during cooling run on the DSC trace. The solid triangles are the phase separation temperatures at various compositions obtained from cloud point measurements...... 51

xiii 2.11 Experimental phase diagram of the sPP/EPDM blend determined by Ramanujam and Kyu [55] using a combination of DSC and light scattering techniques, exhibiting the combined LCST (solid circles) and UCST (solid triangles) together with the melting point depression (open and solid squares). L1 +L2 represents the LCST miscibility gap. The inverted solid triangle symbols are the experimentally observed DSC peaks during cooling run. The authors suggest that S1 + L2 indicates the presence of coexistence of a solid phase with two liquid phases. The solid and open symbols at the bottom of the ﬁgure are the recorded glass transition temperatures (Tg) of the blend using DSC. Two distinct Tg can be observed suggestive of the limited miscibility of the blend. The lines are merely drawn to guide the eye...... 53 2.12 Phase diagram of a mixture of polyethylene fractions of two molecular weights 1000 and 2000 showing ideal solid solution behavior as determined by Smith and Manley [47]. The solid circles denote the experimentally observed melting temperature for various composition of the blend. Only a single melting temperature was observed suggestive of an ideal solid solution with a common melting point lying intermediate of the pure component melting temperatures. The solid line is merely to guide the eye...... 55 2.13 Phase diagram of a mixture of PCL trioxane mixture showing eutectic behavior as determined by Wittman and Manley [46]. The open circles are the melting temperature of the blend obtained from DSC traces. The DSC trace also has an invariant eutectic temperature peak which has is represented by the open triangle. The eutectic peak had a small unknown shoulder at a slightly higher temperature that is represented by the open squares...... 57 2.14 Spherulitic growth of PVDF with heterogeneous nucleation. This is an example to illustrate the growth of an α crystalline phase growing in a β amorphous phase. The PVDF was supercooled below its melting temperature by around 5oC and snapshots were taken at 1 minute intervals after a induction time of 12 mins ...... 58 2.15 From top left to bottom right we observe diamond shaped, lozenge shaped curved, lenticular curved, and slender curved single crystal that have been predicted by Mehta and Kyu [34]...... 63

xiv 2.16 Usually the crystalline growth front in 2D is a ﬂat crystal-melt interface. Atomic scale roughness or the presence of impurities can lead to a roughening of this interface. Under such circumstances, the heat liberated due to crystallization can get trapped locally in the valleys of the rough interface. Xu et al. [57] have shown that this can lead to intricate pattern formation such as the dendritic morphology and dense lamellar branching...... 65

2.17 Xu et al. [57] have determined a morphology map using the material parameters of isotactic polystyrene. The ﬁgure shows the various possible crystalline structures that can be generated upon varying the supercooling and the extent of surface anisotropy parameter for a six fold anistropic crystal. For small undercoolings (bottom two rows), we observe the formation of a spherulite growth with very little or no internal structure (far left) and faceted single crystal (far right). For larger under coolings (Top two rows), we observe spherulite growth with seaweed and dense lamellar branching morphologies on the far left and the formation of dendrite morphology on the far right...... 66 2.18 Images on the left show the progression of binodal phase separation and the images on the right are their Fourier transforms which can be used to determine the size of the phase separated domains...... 69 2.19 Images on the left show the progression of spinodal phase separation and the images on the right are their Fourier transforms, which can be used to determine the size of the phase separated domains...... 71

2.20 Snapshot of a iPP spherulite growing in a 90/10 iPP-aPP blend superimposed with the concentration profiles of the aPP component along the center line. This experiment was conducted by Billingham et al. [65] using UV fluorescence spectroscopy. The a-PP fraction was modified with a marker molecule that is UV fluoroscence active to monitor the concentration of the atactic fraction within and outside the spherulites of iPP...... 76

xv 3.1 Free energy landscape representing (a) a hypothetical miscible binary mixture showing a single minimum on the front surface (i.e. φ at ψ = 0,) and the dual minima on the side surface on the right (i.e., ψ at φ = 1) and (b) a hypothetical partially miscible binary mixture exhibiting dual minima on both the front surface (i.e., φ at ψ = 0) and the side surface on the right (i.e.,ψ at φ = 1). The solid dots indicate the roots of ∂f/∂ψ = 0. The thermodynamic parameters used are W = 10, ζ = 0.1, ζ0 = 1, r1 = 1 and r2 = 1. a) χaa = 0 b) χaa =3...... 87 3.2 Effect of increasing crystalline-amorphous interaction parameter χca= (a) 0.01 (b) 0.1 (c) 0.3 at 500K. As χca increases the solidus and liquidus lines move downward and the crystal-liquid coexistence region broadens till the solidus line hits the pure component axis. Figure (d) shows the effect of increasing the amorphous-amorphous interaction energy for a finite value of crystalline-amorphous interaction parameter χca = 0.3 at 500K. The liquidus curve shifts downward as the intermolecular interaction of the liquid becomes more attractive. In (a), (b) and (c), χaa was calculated by setting A=3 and TLCST = 600K. In (d) χaa was progressively lowered by changing A=3, 4 and 5 and TLCST = 600K...... 89 3.3 Theoretically calculated phase diagram of PVDF/PMMA blend (indicated by filled dots) and experimental data shown by open diamonds (♦). The experimental data was determined by Nishi and Wang [26]. The calculated liquidus line coincides with the experimentally obtained melting point data. The solidus line is very close to the pure component axis. We denote the coexistence region between the solidus and liquidus lines as Cr+L. The blend exists in the isotropic melt state above the liquidus line. An LCST temperature of 330oC was used in the present calculation. (LCST curve outside scale) 92 3.4 Predicted phase diagrams for crystalline-amorphous mixtures exhibiting (a) isotropic, crystal-liquid, neat crystal region, and the melting transition of the crystallizable component and (b) a tea-pot type phase diagram showing liquid-liquid coexistence region as the upper critical solution temperature envelope protruded above the liquidus line of (a). The L+L region denotes the region of liquid- liquid demixing and the Cr+L region denotes the region of solid- liquid demixing...... 95

xvi 3.5 Comparison between the experimental phase diagram of PCL/PS blend by Tanaka and Nishi (denoted by open diamonds (♦)) and the calculation represented by the solid line. (b) The enlarged melting transition region of the phase diagram showing the fit by the present theory (the filled solid lines) as opposed to the melting point depression (denoted by the crosses (x)) calculated by the original Flory diluent theory using the value of χaa = 1.04...... 99 4.1 The variation of free energy of crystallization as a function of crystal order parameter, ψ, of a pure homopolymer, showing a symmetric double well at equilibrium between ψ = 0 and ψ = 1 representing the melt and the solid phase, respectively. The shape of free energy transforms to asymmetric having the crystal order parameter at the solidification potential less than unity, reflecting the imperfect crystal (i.e., crystallinity of less than one) that may be attributed to the metastable nature of polymer crystallization...... 104 4.2 The variation of the crystal order parameters of the constituents with volume fraction at various temperatures. The parameters uti- c lized where r1 = 10 , r2 = 10 , cω = 0.65 , ∆H1 = 12kJ/mole, c 0 0 ∆H2 = 16kJ/mole,Tm,1 = 500K, and Tm,2 = 480K. The results of the Gauss-Newton minimization show that below the melting temperature the crystal order parameter of both the components are coupled with each other. For temperatures 470 and 450 K, the order parameters slowly decrease as we proceed towards intermediate concentration and sharply falls to zero. This is characteristic of first order transitions and indicates the possibility of crystalline-melt coexistence region. At 440 K, both the order parameters are fully developed suggesting that there is no crystalline-melt coexistence region though the possibility of a crystalline-crystalline coexistence still remains...... 109

4.3 Free energy curves for a binary crystalline eutectic system (left) due to unfavorable solid-solid mixing as a function of descending temperature showing the formation of various coexistence regions such as crystal-melt and crystal-crystal. The parameters are the same as used in Figure 4.2. The corresponding free energy curves show that at 470K exist two crystalline-melt coexistence regions. At 450K, even though we see a crystalline-melt transition in Figure 4.2, the crystalline-melt coexistence might be less favorable than the crystalline-crystalline coexistence. At 440K, it is very clear that the melt state is no longer favorable and a crystalline-crystalline coexistence region exists...... 110

xvii 4.4 Development of a eutectic in an ideal mixture. The solid-solid miscibility parameter becomes unfavorable to mixing and leads to the development of a eutectic: a) ideal crystal solution, cω = 1 ; b) non-ideal solid solution, cω = 0.85 ; and c) eutectic phase diagram at cω = 0.65 . d) When the non-ideality of liquid solution increases beyond the crystal-melt phase transition temperatures, i.e., χaa χcrit , a L+L coexistence curve appears above the Cr+L coexistence curves...... 116

4.5 a) the phase diagram of an ideal liquid solution, but non-ideal solid solution with χaa ≥ χcrit leads to the formation of a peritectic (Cr- Cr-L) and b) the non-ideal liquid solution and non-ideal solid solution with χaa χcrit result in the formation of a monotectic (L-L-Cr) above the peritectic line...... 117

4.6 a) Development of the azeotrope when solid-solid mixing becomes favorable such that the crystal phase is induced in the mixture at a higher temperature than the pure transition temperature of both the components. b) When the liquid-liquid mixing is non-ideal, i.e., χaa χcrit in the vicinity of the transition temperatures results in the formation of a L+L coexistence curve above the Cr+L coexistence curves...... 118

4.7 a) Solid solution of polyethylene fractions in comparison with the experimental data of Smith and Manley [47], b) PCL/trioxane eutectic phase diagram [45] showing the presence of crystal-liquid and crystal-crystal regions bound by the liquidus and solidus lines. . . . . 122 ˜ 5.1 2D simulation of the model with Γφ = 1 andκ ˜φ = 1. The initial concentration of the mixture, φ = 0.9. The thermodynamic parameters are W = 15, ζ = 0.1, ζ0 = 1, χca = 3, r1 = 1, r2 = 1 and χaa = 0. a) ψ ﬁeld shows the formation of the crystalline phase where ψ = 1 is denoted by the white color and the melt is represented by ψ = 0 denoted by the black color (see colorbar) at τ = 40. b) 1 − φ ﬁeld shows the corresponding concentration map of the amorphous material. The darker regions inside the crystalline phase are poor in amorphous content. This amorphous material is excluded from the growing front of the crystalline phase seen as a white halo...... 130

xviii 5.2 1D simulation of the model with the same thermodynamic parameters as used in Fig.5.1. The plots are concentration of the amor- ˜ phous component vs distance for different values of Γφ = (a) 4 (b) 2 (c) 1 (d) 0.5 (e) 0.25 (f) 0.125 at time τ = 4. The crystalline phase was nucleated in the center of the grid. As the crystalline phase grows outward the amorphous component is rejected out as seen by the peaks in the concentration profiles. This region also corresponds to the crystalline phase and melt boundary (not shown ˜ in figure). Γφ 1 implies that redistribution of material is much ˜ faster than the crystalline phase growth rate. As Γφ decreases from (a) 4 to (b) 2, the rejection peak increases and the crystalline phase growth front has moved a smaller distance. Also the concentration of amorphous constituent inside the crystalline phase has increases. ˜ This trend continues in (c), (d) and (e). In (f) Γφ 1, the crystal growth front moves much more rapidly and the rejection peak is quite small as well as very little change between the concentration of the crystalline phase and the melt...... 132 ˜ 5.3 Growth rate at different Γφ = 0, 0.125, 0.25, 1, 4, showing the transition from linear to nonlinear growth rates using the same thermodynamic parameters as used in Fig.5.1. Crystallinity is computed by integration of the ψ parameter over the entire volume and then nor- R R malizing it using the maximum possible value ( ψdV/ ζ0dV ). If ˜ Γφ = 0 no redistribution of amorphous component is possible. This growth rate is linear and similar to that observed for pure crystals ˜ growing from their melts. As Γφ = 0.125 increases the growth rate ˜ becomes slower and nonlinear. It is slowest at about Γφ = 0.25 and then increases again as the redistribution becomes much faster than crystalline phase growth rate...... 133 6.1 Hypothetical phase diagrams for crystalline-amorphous polymer blends exhibiting a combined LCST and UCST intersected by the crystal-melt transition gap bound by the liquidus and solidus lines, showing (a) effect of melting temperature of the pure crystal component increasing from top to bottom (182, 223, and 262oC) and (b) the effect of repulsive crystalline-amorphous interaction energy increasing from left to right. χca parameter is varied from 0.01, 0.1, and 1 at the melting temperature by setting the UCST temperature to 182oC and the LCST temperature to 262oC. The melting enthalpy of the constituent crystal was taken as 1500 cal/mole...... 145

xix 6.2 Comparison between the calculated coexistence line and the cloud points (ﬁlled triangle) of iPP/EPDM and the melting points (ﬁlled circles). The solidus and liquidus lines are virtually overlapped (dots), but the existence of both lines is manifested by the kink in the LCST coexistence line. The phase diagram was calculated us- o ing the material parameters, ∆HiP P = 2110 cal/mol, Tm = 162.5 C, riP P = 1800, rEPDM = 1000 and χca = 0.8 at melting temperature . . . 148 6.3 Comparison between the calculated coexistence curves and experimentally observed LCST (open diamonds) and UCST spinodal gap (open diamonds) showing the liquidus (denoted by open circles) and solidus lines on the pure sPP axis. The material parameters utilized were the enthalpy of fusion of sPP ∆HsP P = 1912 cal/mole, Tm = o 123 C, rsP P = 1800, rEPDM = 1000 and χca = 0.8 at the melting temperature...... 149

6.4 Temporal evolution of the crystalline microstructure in the 50/50 iPP/EPDM blend, following a T-quench from the isotropic melt to a supercooled temperature below both the UCST spinodal gap, showing the growth of spherulitic front in the concentration ﬁeld. However, the over-growth of this spherulitic boundary on the bi- continuous SD domain structures can be seen clearly only in the enlarged version...... 155

6.5 Morphology development in a 50/50 iPP/EPDM during cooling from 230oC to room temperature at a slow cooling rate of 0.5oC/min was studied by Chen et al. [54]. a) Polarized optical micrograph under cross polarizers clearly showing the maltese cross pattern indicative of the spherulite structure. b) Four-lobe clover leaf pattern in SALS in the Hv configuration confirming the existence of the spherulite texture. c) Polarized optical micrographs under parallel polarizers showing the phase separated morphology and d) the ring pattern in the Vv configuration dominated by the concentration fluctuations of the phase separated domains...... 156

6.6 Optical micrographs obtained for a 50/50 sPP/EPDM blend isothermally quenched at 100oC by Ramanujam et al. [55]. Left column depicts the evolution of phase separation under the unpolarized condition and right column indicates the growth of crystals under the cross-polarized condition...... 158

xx 6.7 Optical micrographs obtained for a 70/30 sPP/EPDM blend isothermally quenched at 100oC by Ramanujan et al. [55]. The feft column depicts the evolution of phase separation under the unpolarized condition, and the right column indicates the growth of crystals under the cross-polarized condition...... 159 6.8 Emerged bi-continuous structure following a temperature jump from the isotropic melt between the LCST and the melting transition into the LCST gap, which is presumably driven by liquid-liquid phase separation through spinodal decomposition in the 50/50 sPP/EPDM mixture: (a) 1000 s and (b) 3000 s ...... 164 6.9 Competition between the liquid-liquid phase separation through spinodal decomposition and the crystalline structure formation in the 50/50 sPP/EPDM blend. The crystallization occurs with the preformed SD networks. The top and bottom rows represent the temporal evolutions of the concentration field and the corresponding crystal order field...... 165 6.10 Competition between the liquid-liquid phase separation through nucleation and growth showing the droplet domains and the crystalline structure formation in the 70/30 sPP/EPDM blend. The crystalline sPP component being the major phase, the sPP crystallization occurs in the matrix by weaving around the EPDM domains. The top and bottom rows represent the temporal evolutions of the concentration field and the corresponding crystal order field...... 166 6.11 Competition between the liquid-liquid phase separation through nucleation and growth and the crystalline structure formation in the 30/70 sPP/EPDM blend. The crystallization of sPP is confined to the sPP-rich droplets. The top and bottom rows represent the temporal evolutions of the concentration field and the corresponding crystal order field...... 167 7.1 Comparison of two dendritic growth simulations. Top is the dendritic growth in a pure component and bottom is the dendritic growth in a crystalline-amorphous blend. The pictures are the surface maps of the crystal order parameter ψ...... 174 A.1 Demonstration of the modified common tangent algorithm to accurately predict phase diagrams even for highly asymmetric mixtures. X1/X2 = 1, 10, 100 from left to right ...... 189

xxi CHAPTER I

INTRODUCTION

Polymers are a unique class of chemical compounds that are abundant in nature and serve as the building blocks of life itself. Societies have used these materials for centuries in their routine activities, yet it has only been in the 20th century that the true nature of these super-molecules been discerned to some extent. In the early

1900’s, the concept of valency and the ability of atoms to form molecules had been well known; however, there was a strong resistance to the notion that large molecules built up of thousands of repeat units could be assembled in reality. Only in recent history has the physical state of these materials been probed in a systematic way using theory and experiment. The middle of the 20th century saw great strides being made in the

ﬁeld of statistical thermodynamics and its application to the behavior of polymers.

The book “Principles of Polymer Chemistry” [1] by Flory serves as an essential primer for all the young graduate students who join the polymer science and engineering ﬁeld in understanding the physical chemistry of polymers. Flory’s successful description of the physical state of polymers [2], their behavior in solutions [3] and subsequent validation through carefully designed experiments earned him the Nobel prize in 1974.

Much of his work had been devoted to the elucidation of properties of polymers in their

1 disordered state, also commonly called the amorphous state by polymer scientists.

Through fundamental ﬁrst principle approaches and using rational assumptions, he has solved many beautiful problems of polymer science purely through analytical methods.

My tryst with polymer science and engineering began in 2001 under the su- pervision of Professor Thein Kyu with the intent of investigating the phenomenon of crystallization and phase separation in polymer blends. This area has been an active area of research for the past ﬁfty years. The area of phase separation and solution thermodynamics of amorphous polymer mixtures has been extensively studied by a number of notable researchers [3, 4]. Also, the kinetics of phase separation of disordered ﬂuid mixtures is explained very elegantly by Cahn and Hilliard [5]. In thermodynamics of crystallization in polymer blends, the observed melting point depression was explained by Flory in the 1960’s on the basis of the entropy contributions of chain length and the interaction strength among the polymer solvent pair. Flory laid out compelling arguments to justify the assumptions that led to a simple relation to calculate the melting point depression. One of these assumptions is the supposition that the diluent is restricted solely to the amorphous phase of the material. This was a simplistic assumption and admitted to be suspect by Flory himself [6, 7]. There exist in the literature various instances where the simple analytical equations derived by Flory have been used to determine the interaction strengths between a polymer diluent pair even though they sometimes lead to ambiguous results. Various authors

2 and texts have pointed out this discrepancy and cautioned about the use of the equations derived by Flory [8]. Over the course of the past 50 years, the description of phase transitions in liquid crystalline materials using statistical thermodynamics and the concept of order parameters by researchers such as Maeir and Saupe [9, 10] and

McMillan [11] have proven conclusively that the assumption of the diluent solely in the disordered phase of the blend in such systems is invalid. This prompted us to revisit the ﬁeld of crystalline polymer blends. As recently as 2006, Matsuyama [12] also arrived at a similar conclusion for the speciﬁc case of body centered cubic (BCC) and face centered cubic (FCC) ordering of crystals in a binary solution.

The spatial and temporal growth of crystals from melts and blends has also been investigated in great detail. The contributions of Lauritzen and Hoﬀman [13–18],

Keith and Padden [19, 20], Mansfield [21] and Avraami to establish the kinetics of polymer crystallization have added a great deal of knowledge in the field of polymer crystallization. Yet several key questions remain unanswered, including the effect of temperature, concentration and the role of metastable states that lead to the formation of interesting morphologies such as dendrite, dense lamellar branching and spiral domain wall growth in polymer crystallization. This work tries to resolve some of the unexplained phenomena such as structure evolution during concurrent phase separation and crystallization in polymer blends. This work is an offspring and a continuation of the work of two former colleagues, Dr. R. Mehta [22] and Dr. H.

Xu [23], who have shed light on the spatial and temporal dynamics of polymer crys-

3 tallization and the morphologies emerging due to temperature eﬀects and mechanical deformation at the crystal growth fronts.

The arrangement of this dissertation is as follows. Chapter 2 is an overview of the basic thermodynamics and kinetics of phase separation in mixtures and crystallization of polymers. It critically reviews the work that has accumulated over the past 50 years in this ﬁeld and outlines some of the problems that remain unsolved.

In Chapter 3, the thermodynamics of polymer crystallization using order parameters has been extended to a binary blend comprising of a crystalline and an amorphous component. Existing experimentally obtained phase diagrams of polymer blends are computed using the present model, and the discrepancy in the determination of the

Flory-Huggins interaction parameter has been discussed in the context of the present model. In Chapter 4, this formalism is extended to the case of binary crystalline blends to describe ideal and eutectic crystalline mixtures. This extended model is used to calculate existing experimentally obtained phase diagrams of binary crystalline polymer blends. The azeotrope and peritectic phase diagram topologies has also been predicted. The equations of motion to study the spatial and temporal dynamics of crystallization in polymer blends are developed in Chapter 5 using the

Ginzburg-Landau approach. The interplay of thermodynamic and kinetic factors involved in crystallization in polymer blends has been investigated using numerical simulations. The rejection of an amorphous component from a growing crystalline phase and the distribution of the amorphous component within the crystalline phase have

4 been elucidated from the results of the simulations. Chapter 6 is an application of the present thermodynamic and kinetic model to elucidate the phase behavior of thermoplastic elastomer blends and explain the formation of the unique bi-continuous crystalline morphologies that are known to impart high toughness and impact strength to such blends. The dissertation ﬁnishes with conclusions and recommendations for future research in this ﬁeld.

5 CHAPTER II

LITERATURE REVIEW

2.1 Basic Thermodynamic Relations

Equilibrium is deﬁned as the state of a system in which the properties do not change with time, state of rest or balance due to the equal action of opposing forces [24]. Understanding the equilibrium thermodynamics of a multi-component system is a prerequisite to studying the dynamics of phase separation in liquid-liquid and solid-liquid mixtures. All materials are dictated by the universal dictum of moving towards a state of minimum free energy.

2.1.1 Closed Systems

The ﬁrst law of thermodynamics can be expressed such that for a closed system consisting of n moles

d(nU) = T d(nS) − P d(nV ) (2.1) where U, S, and V are molar values of the internal energy, entropy and volume, respectively. The enthalpy (H) and the Gibbs free energy (G) are deﬁned by the following relations

6 H = U + PV (2.2a)

G = H − TS (2.2b)

By taking the total diﬀerential of Equations (2.2b) and (2.2a) and substituting in

Equation (2.1) yields the following relation

d(nG) = (nV )dP − (nS)dT (2.3)

In the case of a single-phase ﬂuid, the system is of constant composition and certain thermodynamic relations can be deduced for isothermal and isobaric cases based on

Equation (2.3)

∂(nG) = nV (2.4a) ∂P T,n

∂(nG) = −nS (2.4b) ∂T P,n

2.1.2 Open Systems

For an open system, in which matter can be exchanged with its surroundings, the Gibbs free energy becomes a function of not only temperature and pressure but also the number of moles of each chemical species present.

∂(nG) ∂(nG) ∂(nG) d(nG) = dP + dT + dn (2.5) ∂P T,n ∂T P,n ∂n T,P

7 The derivative of the Gibbs free energy with respect to the number of moles of species i at constant pressure, temperature and constant number of moles of other species j is called the chemical potential of species i

∂(nG) µi = (2.6) ∂n i T,p,nj

Thus the total diﬀerential of the Gibbs free energy for a multi-component system takes the following form

X d(nG) = (nV )dP − (nS)dT + µidni (2.7) i

When there are two phases in a closed system, α and β, in equilibrium at constant temperature and pressure, the total Gibbs free energy of mixing can be expressed as

m X α α X β β d(G ) = µi dni + µi dni = 0 (2.8) i i

α From the conservation of mass in the system we can deduce the relation dni =

β −dni . Hence, at constant temperature and pressure, for an open system the following relation should hold true at equilibrium.

m α β α d(G ) = (µi − µi )dni = 0 (2.9)

α Since dni is independent and arbitrary, the right-hand side of Equation (2.9) can be zero only if the term in parentheses is zero resulting in the following relationship.

8 α β µi = µi (2.10)

This relation provides us a very interesting insight on the coexistence of phases, and the following postulate can be proposed. For two phases to be in equilibrium with each other, chemical potential in each phase with respect to each species should be equal.

In order to determine the chemical potential, we need to have an accurate description of the free energy in terms of the thermodynamic variables. The total free energy of mixing is the sum of the enthalpy and entropy contributions, ∆Gm = ∆Hm − T ∆Sm.

Four types of mixtures can be identiﬁed on this basis.

1. Ideal solution : ∆Hm is zero and ∆Sm has the ideal form, therefore ∆Gm is negative for all compositions and mixing always occurs.

2. Athermal solutions : ∆Hm is zero and ∆Sm deviates from the ideal form.

3. Regular solutions : ∆Hm is ﬁnite but has the ideal form

4. Irregular solutions: Both ∆Hm and ∆Sm deviate from ideality.

Polymer solutions and blends exhibit large deviations from the characteristics of ideal solutions and belong to the class of irregular solutions. In the proceeding sections, we shall evaluate the free energy of crystallization of a homopolymer, the free energy of mixing of amorphous polymer blends, and the free energy of crystallization in crystalline polymer blends.

9 2.2 Classical Thermodynamics of Crystallization

Polymers are a unique class of materials that rarely reach equilibrium during crystallization due to the long chain nature of the molecules. It is helpful to understand first the equilibrium theory of crystallization and the subsequent modifications that are able to accommodate the peculiar crystallization habits of polymers such as lamellar thickening and metastable states arising due to non-equilibrium effects.

2.2.1 Equilibrium Theory of Crystallization

A polymer molecule will try to minimize its free energy at all times. If a molecule is able to arrive at its minimum energy state, such a molecule is said to be in equilibrium. Thus, in order to study the equilibrium of a polymer molecule undergoing crystallization, it is essential to obtain the Gibbs free energy of such a molecule. The free energy for a crystal may be divided into two parts comprised of a bulk and a surface contribution. Below the melting point of the crystal the bulk free energy of the crystal (∆F ), defined as the difference between the volumetric free energy of a crystal (Fc) and the melt (Fa), is negative. Surface tension is defined as the amount of work required to increase the surface area by one unit, and thus for a crystal surface surface tension is equivalent to the excess surface free energy of the crystal (σ). This free energy has the unit of energy per unit area and can vary for different growth surfaces of a crystal. Assuming chain folding occurs on only one

10 surface, for a crystal of average thickness (l) and fold surface area (A), the Gibbs free energy is given as

c X ∆G = −Al∆F + aiσi + 2Aσe (2.11)

where ∆F = Fc −Fa, ai are areas of diﬀerent growth planes and σe and σi are surface free energies for the folded surface and these lateral growth planes, respectively. At all

0 temperatures below the melting point of the crystal (Tm), bulk free energy change is always negative. If the surface area of the fold plane is very large and the surface free energy of the folded surface is larger than the surface free energy of the other surfaces, then we may neglect the contribution of free energy by the growth fronts. This is a rational argument in the case of polymer molecules since polymer folding should lead to excess in the stress on the fold surfaces and we can neglect the contribution of the second term on the right-hand side in Equation (2.11). Therefore, the stable fold

0 period l would be inﬁnite at all T < Tm. This implies that the extended chain crystal is the most stable form of the crystal. Such a structure is not commonly found in the case of polymers, which means that the more probable structure is a metastable state that is at least more stable than the melt. In order to estimate the lamellar thickness at this metastable state, we ﬁrst calculate the minimum lamellar thickness

(lmin) at which the crystal would be more stable than the melt.

2σ l = e (2.12) min ∆F

11 Thus at temperatures, T < Tm, the lamellar thickness follows the relation, l >

2σe/∆F . In order to obtain more information about this thickness we need a better description of ∆F . The bulk free energy of the crystal must be a function of crystallization temperature, ∆F (T ) = ∆H(T ) − T ∆S(T ), the enthalpy and entropy of the crystal. At the equilibrium melting point we can set this relation to zero,

0 0 0 ∆F (Tm) = ∆H(Tm) − T ∆S(Tm) which gives us a relation for the entropy of the

0 0 0 system, ∆S(Tm) = ∆H(Tm)/Tm. We deﬁne the quantity supercooling (∆T ) as the diﬀerence between the temperature of the material and its equilibrium melting point,

0 ∆T = Tm − T . For small supercooling, the enthalpy and entropy of crystallization can be assumed to be independent of the temperature. Thus, under equilibrium conditions, 0 0 0 ∆H(Tm) ∆F (Tm) = ∆H(Tm) − T 0 (2.13a) Tm

0 0 ∆T ∆F (Tm) = ∆H(Tm) 0 (2.13b) Tm

This provides more information about the minimum thickness of the crystal. On substitution of Equation (2.13) in Equation (2.12), we get

0 2σeTm lmin = 0 (2.14) ∆H(Tm)∆T which shows that the minimum thickness of the lamella is inversely proportional to the supercooling. This diverges to inﬁnity as the supercooling goes to zero which

12 conﬁrms that the true equilibrium structure will be an extended chain crystal but under ﬁnite values of supercooling several metastable states can exist.

2.2.2 Gibbs-Thompson Equation

0 For a given temperature quench, Tc < Tm, we can now determine the Gibbs free energy using Equation (2.13). The lamellar thickness (l) is greater than lmin deﬁnitely by some unknown quantity δl. Using the same principles as described before, ∆F = 0 at the melting temperature of the lamellae Tm. From Equation

(2.11), we obtain the following relation.

0 Tm − Tm ∆G = Al∆H 0 + 2Aσe = 0 (2.15) Tm

Thus, a relation for the melting temperature (Tm) and the thickness (l) of lamella is obtained 2σ T = T 0 1 − e (2.16) m m l∆H

Using these relations we can determine the values for the equilibrium melting point and the fold surface free energy from experimental determination of lamellar thickness

(l) as function of Tm. A plot of Tm versus 1/l under the linear ﬁt approximation as described by Equation (2.16) would yield the equilibrium melting point through the y-intercept and the fold surface free energy from the slope.

13 2.2.3 Week-Hoﬀman Plots

The direct measurement of the lamellar thickness can be quite laborious and prone to error, even though advances in microscopy techniques such as Atomic Force

Microscopy (AFM) and Transmission Electron Microscopy (TEM) have made such measurements possible. Another relation obtained from the aforementioned thermodynamics is the Week-Hoﬀman’s relation. The relation l = lmin + δl can be rewritten

0 as l = βlmin where β always greater than 1 for Tc < Tm. Using this relation in

Equation (2.16) yields 0 2σe Tm = Tm 1 − (2.17) lminβ∆H

Upon substitution of Equation (2.12) in Equation (2.17), we rewrite the relation

1 1 T = T + 1 − T 0 (2.18) m β c β m

0 This relation can yield us the equilibrium melting temperature (Tm) and the β factor upon experimental determination of Tm and Tc.

2.3 Flory-Huggin Theory

In this section, we review the derivation of the thermodynamics of solutions of polymers. This theory was formulated independently by P. J. Flory and M. L.

Huggin in the middle of the last century. The entropy and enthalpy of mixing of two polymers was derived and phase diagrams of polymer pairs were explained on the basis

14 of enthalpy interactions. We will also review contributions of several researchers that have extended this theory to elucidate a host of interesting phase diagrams that occur in the literature.

2.3.1 Entropy of Mixing of Simple Molecules

For a two component system containing simple non-polar liquids, the entropy of mixing (∆Sm) is deﬁned as

∆Sm = −kB(N1lnx1 + N2lnx2) (2.19)

where Ni represents the number of molecules of species i, xi is the mole fraction of species of i for a multi-component mixture, and kB is the Boltzmann constant. The mole fraction of species i is further deﬁned as

Ni xi = , i 6= j (2.20) Ni + Nj

2.3.2 Entropy of Mixing of Polymers

A polymer chain molecule in itself is a large and convoluted structure of repeat units and can assume a high number of configurations by itself. Entropy is defined as the degree of randomness of a system. Thus, a polymer chain molecule has a higher entropy than an ordinary small molecule. Consequently, two polymer molecules in a blend are not affected drastically by mixing as the increase in entropy

15 due to mixing is marginal. The determination of the entropy of mixing for binary polymer blends is important for the rigor of the theory and is deﬁned analogous to simple liquids.

∆Sm = −kB(N1lnφ1 + N2lnφ2) (2.21)

where φi is now the volume fraction of component i which is further deﬁned on the basis of the number of sites each polymer species occupies in a lattice, or alternatively speaking, the chain length (Xi). Using the constant density approximation yields a correlation between mole fraction and volume fraction in a straight forward manner.

NiXi φi = , i 6= j (2.22) NiXi + NjXj

Substituting Equation (2.22) in Equation (2.21) yields an expression for the entropy of mixing for two polymers accounting for chain length [3,4].

φ1 φ2 ∆Sm = −kB( lnφ1 + lnφ2) (2.23) X1 X2

2.3.3 Enthalpy of Mixing of Polymers

According to Flory’s approach [3], the enthalpy of mixing (∆Hm) for a polymer solution or blend consisting of two components can be calculated by taking the diﬀerence between the enthalpy of the solution or blend (H12) and the enthalpy of the pure components (H11 and H22) to yield the relation ∆Hm = H12 − (H11 + H22).

16 H11, H22, and H12 are deﬁned using an interaction energy (ij) that exists between every two segments, each contributing ij/2 to yield the relations

1 1 H = N X z = Nφ z (2.24a) 11 1 1 2 11 1 2 11

1 1 H = N X z = Nφ z (2.24b) 22 2 2 2 22 2 2 22

1 1 1 1 H = N X z φ + φ + N X z φ + φ (2.24c) 12 1 1 2 11 1 2 12 2 2 2 2 22 2 2 12 1

where N is the total number of lattice sites equal to N1X1 +N2X2 for a binary system and z is the number of neighbors to one segment, also called the coordination number.

In the case of a complete random packing of the mixture, this coordination number z is usually taken as 10 but can vary from 4 − 12. Hence, the total enthalpy of mixing is derived as 1 1 ∆H = zNφ φ − − = zNφ φ ∆ (2.25) m 1 2 12 2 11 2 22 1 2

∆ represents the average energy due to contact of segments. In a solvent/polymer solution, z neighbors surround each solvent unit, the Flory-Huggins interaction parameter can be introduced as a dimensionless quantity

z∆ χFH = (2.26) kBT

The initial theory was devised for a polymer solvent system, and it was assumed that all the lattice sites are occupied with either of the two components. For the case

17 of a blend containing two polymers, a variety of effects would have to be taken into account, such as incomplete filling of the lattice sites, chain connectivity, branching and more. It is for this reason that the Flory-Huggins interaction parameter is not generally calculated by this expression. An empirical relationship is used to define the parameter that has a reciprocal dependence on absolute temperature as described by the theory and whose constants can be derived and fitted to the experimental phase diagrams to account for the deviations in both the entropy and enthalpy of mixing in real polymer mixtures [4].

χ = A + B/T (2.27) where A and B are constants. The Flory-Huggins interaction parameter is directly related to the critical temperature (Tcrit) and the interaction parameter at the critical point (χcrit) (see Appendix B) that can be calculated by Scotts relation [1].

√ √ 2 ( X1 + X2) χcrit = (2.28) 2X1X2

Several empirical modiﬁcations to the Flory-Huggins interaction parameter have been established in the literature. Koningsveld [8] expressed a modiﬁed χ(T, φ) as a product of a temperature contribution multiplied by a concentration contribution

B χ = (A + + ClnT )(1 + Dφ + Eφ2) (2.29) T

18 This form of χ is used to capture all the non-combinatorial entropy eﬀects and enthalpy eﬀects by just one term rather than adding more terms to account for every non-ideality of the system.

2.4 Determination of Phase Equilibria

In this section we review the methods of calculating phase diagrams for amorphous polymer blends that exhibit liquid-liquid phase separation behavior. The sig- niﬁcance of the binodal and spinodal regions in the phase diagrams exhibiting liquid- liquid phase separation is explained, followed by the calculation of phase diagrams of crystalline polymer blends that exhibit a solid-liquid phase separation behavior using the Flory diluent theory.

2.4.1 Gibbs Criteria

The incompressibility assumption, φ1 + φ2 = 1, leads to the reduction of the free energy (∆Gm) in a single independent thermodynamic variable φ1 ≡ φ and

φ2 ≡ 1 − φ. Gibbs derived a necessary condition for the stability of a ﬂuid phase that the chemical potential of a component must increase with increasing density of that component. In the context of a two component system this yields the relation,

∂2∆G m ≥ 0 (2.30) ∂φ2

19 0

−0.05 225

−0.1 250

−0.15 275 ) φ f(

−0.2 300

−0.25 325

T=350 K −0.3 T =300K, X =1, X =1 crit 1 2

−0.35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 φ

Figure 2.1: Free energy of mixing of a hypothetical binary liquid mixture with A = 0, X1 = 1 and X2 = 1 with Tcrit = 300K has been plotted as a function of composition for various temperatures to illustrate the appearance of a double well structure below the critical temperature that leads to UCST type liquid-liquid phase separation. For temperatures above Tcrit (350K and 325K) the free energy is a single well and the mixture is stable. As temperature decreases below 300K a local maximum at φ = 0.5 is discerned that increases with decreasing temperature. The local maximum indicates that the mixture is unstable according to the Gibbs criterion and the equilibrium compositions of the phases can be found by the common tangent method. The dotted line plotted for the free energy of mixing at T = 225K is the common tangent line and the points at which it touches the free energy curve are the equilibrium compositions.

20 If this condition is not satisfied, then the mixture becomes unstable with respect to any infinitely small composition fluctuations. Also, the total free energy of mixing should be negative for the process to be thermodynamically favorable, i.e.

∆Gm < 0. It is a fundamental principle of thermodynamics that materials always progress towards a state of minimum free energy until equilibrium is reached. This concept is expressed mathematically utilizing the Gibbs free energy of isotropic mixing. For a binary system in equilibrium, the chemical potentials of each component in each phase must be the same as in the other phase. We assign these phases to be

α and β. In order for stability to occur in a two-phase system, the chemical potential

α β with respect to each component should be balanced in both the phases, µ1 = µ1 and

α β µ2 = µ2 .

2.4.2 Liquid-Liquid Mixtures

We rewrite the complete Flory-Huggin’s expression for the mixing of two polymers upon applying the incompressibility assumption and renormalization. This dimensionless free energy, gl also referred to as the local free energy density can be written as

∆G φ (1 − φ) gl = m = ln φ + ln(1 − φ) + χφ(1 − φ) (2.31) kBT X1 X2

At constant density, due to the nature of the Flory Huggins interaction parameter, the free energy curve can exhibit a double well structure such that multiple phases

21 can coexist with each other (Figure 2.1). A complete map of the coexistence boundaries in the composition space as a function of temperature is called a phase diagram.

The region of instability that is encompassed by the coexistence boundaries is also known as the binodal curve. As shown in Figure 2.2 within the binodal curve, the region is unstable and is further subdivided into a metastable region where Equation

(2.30) is satisfied and a critically unstable region where the border is termed as the spinodal. Within the metastable region, the system is stable against small concentration fluctuations. In the spinodal region, even the smallest concentration fluctuations can grow.

2.4.2.1 Spinodal Region and Critical Points

The inflection points of the free energy curve in Figure 2.1, otherwise known as spinodal points, define the limits of thermodynamic instability against any concentration fluctuation. In mathematical terms, the spinodal points are determined by taking the second partial derivatives of the free energy of mixing with respect to

φ and equating to zero and can be determined analytically.

∂2∆G m = 0 (2.32) ∂φ2

The critical point on the phase diagram [1] can be determined from applying the third partial derivative of the free energy of mixing with respect to φ. Equating it to zero, we obtain

22 Figure 2.2: Phase diagram exhibiting an upper critical solution temperature has been calculated using the Flory Huggins theory of isotropic mixing to illustrate the binodal (dark black line) and the spinodal regions (gray line) using the parameters A = 0, X1 = 1 and X2 = 1 with Tcrit = 500K. The mixture exists as a single phase outside the binodal envelope and is termed stable. The equilibrium compositions that determine the binodal envelope are calculated using the common tangent approach. Additionally a spinodal envelope is calculated within the binodal envelope by setting the Gibbs inequality in Equation (2.30) to zero. The region between the binodal and spinodal envelope is termed metastable

23 ∂3∆G m = 0 (2.33) ∂φ3

Solving this equality, the critical volume fraction φcrit and the critical interaction parameter χcrit can be determined analytically as

√ √ 2 ( X1 + X2) χcrit = (2.34a) 2X1X2

√ X2 φcrit = √ √ (2.34b) X1 + X2

2.4.2.2 Binodal Region: Michelson’s Method

The binodal points which define the complete region of instability are much trickier to determine analytically and are usually determined using numerical algorithms. Michelson’s method of determining the binodal curve, which is a modification of the tangent plane minimization approach [25], is one of the methods used to compute multi-component fluid phase equilibria. The tangent plane minimization approach to find the global minimum can be outlined as follows. An objective function is defined which represents the distance of a tangent plane of the Gibbs free energy surface at a particular concentration at any other concentration. For a binary system the plane reduces to a line to the Gibbs free energy curve. This is shown in the Figure 2.3.

24 Figure 2.3: Michelson’s method to ﬁnd the binodal points on the phase diagram using the tangent plane minimization approach [25]. For a binary system, this method uses a probe concentration φz and uses a slope matching algorithm to ﬁnd concentrations φs1 and φs2. If the probe concentration is within the binodal then it is never the lowest tangent line, D(φs1, φz) < 0 or D(φs2, φz) < 0. These concentrations of the liquid mixture are unstable. If both are greater than zero, then the mixture is stable. By mapping this transition from stable to unstable we can determine the equilibrium composition (φα and φβ) of the two phases.

25 n X ∂∆Gm D(φ, z) = ∆G (φ) − ∆G (z) + (φ − z ) (2.35) m m ∂φ i i i=1 φ=z

Here z represents the location of an arbitrary volume fraction φz. This objective function needs to be minimized with respect to each constituent’s volume fraction, and for a two component system implies ﬁnding the tangent to the free energy curve.

From the ﬁgure we can see that at φz we have a trivial solution, and at φs1 and φs2 the objective functions are positive and negative respectively. Michelson observed that the global minimum is not at φz if a negative value of the objective function exists that is below the tangent line at z, thus causing feed at composition to be unstable and to phase separate. The feed is stable when both the roots for the objective function are positive. We can run this procedure for all compositions sequentially from 0 to 1 and note the transition where the feed turns from stable to unstable and vice versa. The computer algorithm is fairly easy in description though highly computationally intensive since for each composition a slope matching procedure needs to be run which makes the problem a N 2 type computational problem. We run this procedure for various temperatures at constant pressure to construct a temperature versus composition diagram. Using various χ interaction parameters, a host of phase diagrams found in real systems can be calculated.

26 2.4.3 Solid-Liquid Mixtures

In this section, we review the work of Prigogine [8] and Flory [6] to explain the depression of melting points of molecules in solution. Prigogine explained the phenomenon on the basis of entropy of mixing of the molecules and later Flory extended it for non-ideal polymer mixtures and elucidated the contribution of both the entropy and enthalpy eﬀects of mixing to the observed depression of melting point.

2.4.3.1 Prigogine’s Model

Prigogine et al. [8] were the ﬁrst to establish phase diagrams for crystal-liquid transitions in binary mixtures theoretically by equating the solid and liquid chemical potentials under a set of assumptions: (1) solvent and solute are, like molecules indistinguishable from each other, (2) the liquid mixture is completely ideal, and (3) the solvent is completely immiscible in the solute phase. These assumptions appeared to hold for describing a variety of phase diagrams of small molecule systems such as o-chloronitrobenzene and p-chloronitrobenzene mixture and the mixtures of aro- matic hydrocarbons in benzene. According to this set of assumptions we can directly evaluate the melting point depression by equating the chemical potential of the solid phase to the known chemical potential of the liquid phase. If free energy change upon crystallization is assumed to be a colligative property in solutions, the chemical potential of the solid crystalline fraction (µc) will be equal to the corresponding

27 intensive property, i.e. free energy as calculated in Equation (2.13).

0 ∆Hu Tm − Tm µc = 0 (2.36) kBTm Tm which leads us to the expression,

1 1 −R − 0 = lnφ (2.37) Tm Tm ∆Hu where R is the universal gas constant. This model relies simply on the entropy energy of a mixture. The model is overly simplified and does not account for the non-ideal nature that can exist in the liquid phase due to enthalpy interactions. Non-ideality of the liquid phase has been observed in the mixtures of hydrocarbons in carbon- tetrachloride solutions that cannot be explained by this model. Another lacuna exists in the entropy part of the free energy which is unable to describe materials like polymers and long hydrocarbons that contain a large number of monomeric units and have large conformational energy. Thus, the model is severely limited to the use of very specific set of mixtures and open to refinement. Flory subsequently applied the Flory-Huggins theory of polymer mixing to accurately model the enthalpy and entropy correlations in polymer solvent systems which is described in the following section.

28 2.4.3.2 Flory Diluent Theory

Blends of polymers in solvents or polymers usually exhibit a reduction in their melting points. This phenomenon is termed as melting point depression. The most widely used theory for calculating the eﬀect of blending on the melting point of polymers is the Flory diluent theory [6]. The theory is based on solving the Gibb’s condition for coexistence such that the chemical potential in the solid and liquid phase be equal. It had been long thought before the establishment of phase diagrams for alloys, nematic mixtures, and smectic mixtures of liquid crystal/solvent blends that the solid phase formed in a blend would be in its pure state. The Flory-diluent theory is also formulated on this premise. This assumption of a pure solid state is also the only way to obtain an analytical solution to Gibb’s coexistence rule. This provides an insight about the widespread adoption of this model in industry and academia as well. It is an extension of the aforementioned Prigogine model with the inclusion of the Flory-Huggins free energy of mixing to describe non-ideal liquid mixtures. Let us consider the case where a crystalline polymer is mixed with another amorphous polymer/solvent. The pseudo chemical potential of the liquid phase can be calculated from the Flory-Huggins theory of liquid - liquid mixing for polymer blends as determined in Equation (2.31) to obtain

∂gi 1 + ln(φ) 1 + ln(1 − φ) = µl = − + χ(1 − 2φ) (2.38) ∂φ X1 X2

29 An alternate method to determine the Gibb’s coexistence condition by equating chemical potentials is to express it as a common tangent problem as described earlier. This geometrical interpretation of the problem renders a set of equations for determining the solid-liquid equilibria according to the Flory diluent theory. Assuming the solid phase, φα = 1 the equation to be solved is as follows

i c i ∂g g − g (φβ) = β (2.39) ∂φ φβ 1 − φ

On substitution of Equations (2.36) and (2.38) and signiﬁcant mathematical jugglery, we can obtain the relation for the melting point depression curve.

1 1 kB ln(φβ) 1 1 2 − 0 = + − (1 − φβ) + χ(1 − φβ) (2.40) Tm Tm ∆Hu X2 X2 X1

For the case of small molecule systems, the melting point depression is a consequence of the high entropy of mixing and follows a linear relationship for small dilutions.

In the case of polymers where the entropy contributions might be negligible, the interaction parameter χ plays an important role in determining the melting curve.

According to the original derivation by Flory, the mixing energy was formulated for non-polar polymer solvent pairs and no speciﬁc interactions such as hydrogen bonding were considered. For this reason the amorphous interaction parameter χ has been modeled by an equation of state model like Van Der Waal’s model to measure the amorphous-amorphous interaction energy.

30 2.4.3.3 Estimation of χ Parameter

The prevailing misconception of determining the χ parameter on the basis of the Flory diluent theory requires further discussion. Upon closer inspection of Equa- tion (2.40), we can observe that for high molecular weight polymer blends, the entropy contributions are negligible. This means that the χ parameter can be determined as a mere function of the thermodynamics of crystallization. This approach was ﬁrst utilized by Nishi and Wang [26] to determine the χ of the PVDF and PMMA blend.

It is not surprising to note that the more accurate the thermodynamics of crystallization are, the better the approximation of χ will be. Even with a highly detailed model of the thermodynamics of crystallization, the assumption of pure solid formation in blend (φα = 1) is irrational. There is ample evidence in the literature especially for ﬁrst order transitions in liquid crystal blends that the ordered phase can form as mixture of the two components at intermediate concentrations under equilibrium conditions. Another lacuna exists for the case of polymers whose χ value might be positive. For these cases the theory might predict an increase in melting point.

This limits the use of this theory to the cases of miscible blends. Burghardt [27] has predicted the melting curves for the cases of polymer solvent mixtures which show χ > χcrit around the melting temperature of the polymer, i.e. a liquid-liquid miscibility gap might exist in the vicinity of the solid-liquid transition temperature.

He probed the particular problem of polymer-solvent mixtures in which the entropy contributions are still prevalent due to the disparity in size of the polymer molecule

31 Figure 2.4: Illustration of the graphical procedure to equate the chemical potential of the solid and liquid phases to determine the phase diagram

32 compared to the solvent. In Figure 2.4, we see the graphical illustration of the common tangent method as applied to the Flory diluent theory. It is quite clear from the ﬁgure that the abberation of melting point elevation for partially immiscible mixtures is a consequence of the mathematical treatment of the problem. It should not occur in reality. In reality, no melting point depression is observed under the binodal envelope and thus for a highly immiscible blend, the depression of the melting point should be virtually negligible. Another artifact of the equations is the formation of the ‘Van Der Waal’s’ loop inside the binodal envelope as shown by the dotted lines.

This is due to the nature of the melting point depression equation which predicts multiple real roots for χ > χcrit.

2.5 Thermodynamics of Crystallization Based on Order Parameter Approach

The physical nature of ﬁrst order transitions with the discontinuities involved in the primary thermodynamics quantities like free energy leads to some rather cumbersome models for the dynamics. One severity of utmost importance is the tracking of the solid liquid interface [28]. This problem is also sometimes referred to as the

Stefan’s sharp boundary problem. For simple scenarios it might be feasible but for studying crystallization under multiple nucleation events in 2 or 3 dimensions, the boundary tracking problem can become a very tedious task computationally. In order to overcome this problem , the concept of a phase field model was first proposed by J.S Langer [29] for the solidification of metals. The first instance of describing

33 the physics of first order transitions using the phase field model should be attributed to Ginzburg and Landau who used this approach to describe the phase transition associated with super-conductivity [30, 31]. In this approach, instead of a sharp dis- continuity in the free energy at the phase transition temperature, a smooth transition over the interface of the transition is assumed. In the case of solidification, an additional thermodynamic variable is defined to characterize the state of the system and describe its free energy. It is called the crystal order parameter (ψ). The thickness of the interface is extremely small but it is finite; hence, the problems of boundary tracking can be eliminated using this approach [32,33].

2.5.1 Crystallization in Small Molecule Systems

The local free energy of crystallization is an asymmetric double well potential for crystal ordering with respect to a crystal order parameter ψ as shown in Figure

2.5 [34]. For small molecule systems, this order parameter is employed to describe solid crystalline phase (ψ = 1) or liquid melt (ψ = 0). The local free energy density, the dimensionless free energy, is expressed in terms of a Landau expansion in ψ as follows F c ζψ2 (1 + ζ)ψ3 ψ4 f c = = W − + (2.41) kBT 2 3 4

In this equation W is a dimensionless quantity that can be related to the activation barrier to the process and ζ is a supercooling dependent parameter. As can be witnessed from Equation (2.41), the reference state is assumed to be the pure melt.

34 Hence the function f c = 0 at ψ = 0. The free energy of crystallization should be related to the equilibrium free energy of crystallization. Thus, ∆F = f c(ψ =

1) − f c(ψ = 0) and using Equations (2.13) and (2.41), we obtain1,

∆Hu T ζ 1 ∆F = 1 − 0 = W − (2.42) nRT Tm 6 12

2 where ∆Hu is the latent heat of crystallization and is a negative quantity since

0 the process is exothermic. For the case of small molecule systems Tm has no special meaning and is the equilibrium melting point of the material, R is the universal gas

0 constant, and T is the absolute temperature of crystallization. When T < Tm , ∆F is negative and the crystalline state is preferred over the melt state. At equilibrium

0 c c 0 0 ∆F (Tm) = f (ψ = 1)|Tm −f (ψ = 0)|Tm = 0. The free energy description is physically valid only between 0 < ψ < 1 and is undeﬁned beyond these limits. The complete

Gibb’s free energy of crystallization should also include the penalty of energy of crystallization due to the creation of surfaces. This is expressed as a nonlocal free energy where the position of the growth surface can be identiﬁed using the ∇ψ vector whose square of the norm is positive only on the solid-liquid interface and zero everywhere else. The surface free energy for creating that surface is determined by a second rank

1In the course of the derivations, we observe the scaling of the free energy to dimensionless forms using either RT or NAkBT energy vales. They are one and the same in the context of these derivations since we are calculating free energy on a per kilo-mole basis of the material or Avogadro number of atoms . These quantities are identical in the SI unit system where R = NAkB 2 The sign of ∆Hu can be explicitly be written negative or the quantity can be negative. The derivations are worked out assuming that it is a negative quantity

35 tensor. The dimensionless free energy is called the interface gradient coeﬃcient (κψ)

2 and is expressed as |κψ∇ψ| /2 which is a scalar quantity. Thus, the complete Gibbs free energy of crystallization over the entire volume using the phase ﬁeld approach is expressed as

c Z 2 3 4 c G ζψ (1 + ζ)ψ ψ 2 g = = W − + + |κψ∇ψ| /2 dV (2.43) kBT V 2 3 4

2.5.2 Relationships Between Model Parameters and Material Properties for Small

Molecule Systems

The excess free energy at the interface which is also the surface free energy may be evaluated using the Cahn approach,

σ Z 1 √ = κψ 2∆F dψ (2.44) nRT 0

0 At T = Tm , the surface free energy can calculated by integrating Equation (2.41) to obtain the relation, σ κ W 1/2 = ψ (2.45) nRT 6 2

The interfacial thickness δ can be calculated using the relation,

1 1/2 δ = κψ c (2.46) 2flocalmax

36 Figure 2.5: Local free energy of crystallization f(ψ) is represented by a Landau expansion in ψ. A fourth order polynomial with coeﬃcients representing a double well structure is chosen. The coeﬃcients are related to the heat of crystallization and the supercooling (∆T ) imposed on the system. ψ = 0 represents the melt 0 and ψ = 1 represents the crystal. At T < Tm, crystal is more stable than melt 0 and at T = Tm free energy of both solid and melt are the same and equal to 0 zero. At T > Tm, the melt is at a lower minimum than the crystal. ζ marks the location of the local maximum and f(ζ) is analogous to the nucleation barrier to crystallization.

37 which we can be solved using Equation (2.41) to obtain

2 1/2 δ = 4κ (2.47) ψ W

Solving Equations (2.47) and (2.45) simultaneously, we can obtain the relations for

W and κψ as σ 1 W = 48 (2.48a) nRT δ

3 σ κ = δ (2.48b) ψ 2 nRT

Also using Equation (2.42) and substituting for W using Equation (2.48), we can obtain the relation for ζ as a function of supercooling ∆T ,

1 ∆Huδ T ζ = − (1 − 0 ) (2.49) 2 Rσ Tm

According to the derivation done by Oxtoby and Harowell [35], we can estimate the mobility term Γψ using the velocity of propagation of density waves in the medium through the relation 3 ν = − √ Γψδ∆F (2.50) 2 2

2.6 Phase Equilibria in Binary Crystalline Blends

The order parameter approach has been successfully used to describe several phase diagrams observed in binary crystalline blends. In this section, we review the

38 work of McFadden and Elder to determine ideal solid solution and eutectic phase diagrams respectively. A uniﬁed description of the various possible phase diagram topologies such as ideal solid solutions, eutectic, peritectic and azeotrope for binary crystalline mixtures is still lacking but the following section provides many clues to resolve this problem.

2.6.1 Ideal Crystalline Blends

We review the approach of McFadden et al. [36,37] to describe the phase ﬁeld model of crystallization to binary alloys which are treated as ideal solid solutions. In contrast to the existing theories for evaluating the melting point depression, several key assumptions have been removed. The ideal solid solution is deﬁned as one in which the two components are indistinguishable from each other except in their melting points and their heats of crystallization and that the components are completely miscible in each other regardless of the phase in which they are present. Thus, the crystallization of the mixture is determined as a function of one crystal order parameter ψ. The free energy of crystallization of each component is a colligative

c property and thus can be added according to the mixture rule g = φ1f1(ψ)+φ2f2(ψ).

Each of the individual landau free energies can be related to the classical equilibrium thermodynamics of crystallization using Equation (2.13). This is adequate to explain ideal binary mixtures like the Cu-Ni alloy [38]. Thus, the free energy of crystallization

39 of a binary blend is represented by Equation (2.51)

4 c ψ 1 + ζ1 3 ζ1 2 g = φW1( − ψ + ψ ) 4 3 2 (2.51) ψ4 1 + ζ ζ + (1 − φ)W ( − 2 ψ3 + 2 ψ2) 2 4 3 2

Here W1 and W2 are dimensionless constants characteristic of the nucleation barrier for crystallization of each component. ζ1 and ζ2 are the parameters related to supercooling with respect to the equilibrium melting point of each component respectively

0 0 (Tm,1 and Tm,2) and their heat of fusions. In order to test the validity of this model we analyze the free energy at the limit of pure component system by setting φ = 1 or

φ = 0. We can immediately see that at the limits the pure component crystallization models are recovered, and this assures the model is valid. Thus, the total free energy of a binary blend is the sum of the free energy due to crystallization and the free energy of isotropic mixing g = gc + gl.

We use a free energy minimization approach to calculate the phase diagrams in which the free energy is ﬁrst minimized with respect to the crystal order parameter to determine the global minimum in ψ using a steepest descent algorithm. [39–42]

∂g =φW (ψ(ψ − ζ )(ψ − 1)) ∂ψ 1 1 (2.52)

+ (1 − φ)W2(ψ(ψ − ζ2)(ψ − 1)) = 0

Subsequent to minimization in ψ we calculate the coexistence region by equating the chemical potentials in each phase according to the Gibbs rule (µα = µβ). Find two

40 points on the free energy minimum curve such that the slope of the line connecting them is equal to the derivative of the free energy with respect to the concentration, i.e the chemical potential, at those points. A variety of common tangent planes can exist on the free energy surface which can yield a liquid liquid (LL — ∂g/∂φ|ψ=0 =

∂g/∂φ|ψ=0), a solid-liquid (SL — ∂g/∂φ|ψ=0 = ∂g/∂φ|ψ=1) or a solid-solid (SS —

∂g/∂φ|ψ=1 = ∂g/∂φ|ψ=1) coexistence region. In this particular case only the solid- liquid coexistence curves exist. This is a much more robust way of determining the phase diagrams since we get solutions for both the solidus and liquidus curve.

2.6.2 Eutectic Systems

Eutectic is deﬁned as the point in the phase diagram where a liquid is in simultaneous equilibrium with two solid phases. Hence, the liquid splits into two solid phases. The Flory diluent theory describes such behavior by using two separate equations to determine the liquidus curves, the intersection of which should yield the eutectic point. The solidus curves in this case will be the pure states. This approach has been used by Manley et al. to explain the eutectic phase diagrams for polyethylene fraction mixtures and polyethyleneoxide-trioxane mixtures [45–47].

The experimental data is ﬁtted with the theory using suitable χ values. In order to realize the melting point depression, the system must be miscible in the melt. For determining the dynamics of eutectic evolution, it is necessary to know the miscibility/immiscibility in the solid phase that drives the solid-solid phase separation. Thus, the thermodynamics established in the context of the Flory diluent theory are inca-

41 Figure 2.6: The ideal solid solution phase diagram of a binary crystalline alloy of Copper (Cu) and Nickel (Ni) as established by McFadden et al. [38]. Zero on the abscissa represents pure Ni and one represents pure Cu. The melting temperatures of pure Cu is 1358K and pure Ni is 1728K. Above 1728K , the blend exists in the melt state at all concentrations. At temperatures lower than the melting temperature of Ni, the equilibrium concentration of the solidus and liquidus phases is calculated using the common tangent algorithm to the free energy described in Equation (2.51). The dotted line denotes the crystal-melt transition where the free energy of crystal and melt are equal.

42 Figure 2.7: Thermodynamics of eutectic phase diagrams using the order parameter approach. The phase diagram has been established as a function of ∆T with concentration order parameter (ς). ς is deﬁned as φ − φ0 where φ0 is the initial concentration of the blend. Various coexistence regions (Solid-Liquid) and (Solid-Solid) were calculated using the free energy described by Elder [43, 44]. Two solid-liquid coexistence regions intersect to form the eutectic where the liquid is in equilibrium with two solid phases. Below the eutectic point lies a region of Solid-Solid immiscibility.

43 pable of describing the morphological evolution of the eutectic microstructure. Elder et al. [43, 44] have been working on solving the morphology evolution of eutectics using the Landau type free energy approach. The free energy described by Elder is the following.

g = −aψ2 + bψ4 + (c∆T − dς2)ψ + eς2 + fς4 (2.53) where a, b, c are the coefficients of the crystallization free energy , e, f are the coefficients of the free energy of mixing and d is the coefficient of the coupling term between the two fields. The model is developed on analogous principles to the McFadden approach with the incorporation of a solid-solid immiscibility parameter represented by d. This model has a large number of indeterminable coefficients but it is able to capture the physics of eutectic growth and also yields consistent solutions to determine the phase diagrams. As can be seen from the Figure 2.7, the phase diagram generated using this free energy is not very realistic and the model is only able to predict the 50-50 eutectic phase diagram. Also the nature of the linear coupling term in ψ introduces fluctuations in the local minimum associated with the melt, which is unphysical. The morphology of the eutectics formed in binary alloys can be described by using this free energy functional and plugging it into the equations of motion which will be described in a later section in more detail.

44 2.7 Experimental Phase Diagrams

In this section, we review some polymer systems whose phase diagrams have been investigated experimentally. We ﬁrst describe systems consisting of a crystallisable component and an amorphous component. The blend of poly(vinylideneﬂuoride)

(PVDF) and poly(methylmethacrylate) (PMMA) is studied ﬁrst as a consequence of the large amount of literature available where PVDF is the crystallisable component and PMMA is the amorphous component. Following that is the polycaprolactone

(PCL) /polystyrene (PS) blend, which shows a more interesting phase diagram due to the intersection of the liquidus of the solid-liquid phase separation curve with the binodal envelope of the liquid-liquid phase separation of the UCST type. Two more additional systems of polypropylene (PP) /ethylene propylene diene monomer

(EPDM) rubber mixtures have been described for systems that exhibit both LCST and UCST type behavior and a melting transition. The isotactic polypropylene (iPP)

/EPDM mixture involves the intersection of the liquidus of the solid-liquid phase separation with the binodal envelope of the liquid-liquid phase separation of the LCST type and the syndiotactic polypropylene (sPP) /EPDM mixture which clearly shows experimentally the UCST and LCST boundaries and the melting transition of sPP between them. Then we proceed to describe some experimental systems involving both components that are crystallisable. This will comprise ﬁrst a description of the

PCL/Trioxane system which is known to form eutectic phase diagrams and subsequently a description of the experimental phase diagrams of mixtures of polyethylene

45 (PE) fractions that have been found to exhibit phase diagrams similar to those of ideal solid solutions such as Cu-Ni alloys.

2.7.1 Poly(vinylideneﬂuoride)/Poly(methylmethacrylate) Blends

A simple system consisting of only one crystallizable constituent is the blend of PVDF and PMMA. This blend consists of the crystallizable component, PVDF, with a melting point of around 165oC. The number average molecular weight is around Mn = 200000. PMMA is an amorphous polymer and has a number average molecular weight around Mn = 40000. The blend has an LCST temperature of about

330oC and is completely miscible [48] below this temperature. In the case of the

PVDF-PMMA blend which is a completely miscible blend, Nishi and Wang [26] have used the Van Der Waals model to calculate χ = BV1/RT where B is the interaction energy density and V1 is the molar volume of PVDF. In this particular scenario a value of −3.6 cal/cm3, which gives χ = −0.31 around the melting temperature, fits their data very well. This highly negative value of the χ parameter is suggestive of some stronger specific interactions [49–51] than pure London dispersion forces. The theory seems to describe their data well, but when we comb the literature for values of the χ from other sources we find a broad distribution of values ranging from −0.05 to −0.6 depending on the test methods [52]. It is also important to note at this point that the mixture is miscible completely in all compositions, but it is not a ideal liquid by any sense due to the large negative value of the χ interaction parameter.

46 Figure 2.8: Melting point depression of PVDF/PMMA blend as function of PMMA volume fraction as determined by Nishi and Wang [26]. Open Sym- bols denote experimental data for the observed melting point depression using Dynamic Scanning Calorimetry (DSC). Solid squares indicate the experimantally determined glass transition temperatures (Tg) which are quite invariant to composition. Single Tg is evidence of the miscible nature of the mixture at all compositions.

47 2.7.2 Poly(caprolactone)/Polystyrene Blends

The second mixture of interest is the PCL/PS blend that will allow us to study the interplay of crystallization rate and phase separation dynamics. The mixture still contains only one crystallisable component, though the liquid phase is no longer ideal. The mixture exhibits an upper critical solution temperature at around 140oC and a melting transition of the PCL constituent at about 60oC. The phase diagram for liquid-liquid mixing is highly skewed due to high asymmetry in the size of the species. The PS has only a molecular weight of 1000 compared to about 33000 for the PCL component. If χ is determined from the UCST curve, then the ﬁt to the melting point depression data is not very accurate. This particular mixture exhibits some unique morphologies due to the interplay of phase separation and crystallization which will be explained later.

2.7.3 Blends of Isotactic Polypropylene and Ethylene Propylene Diene Monomer

Polypropylene may be classified in three isomeric forms based on the tacticity of the methylene side group, viz., isotactic, syndiotactic and atactic. Both isotactic and syndiotactic forms are highly crystalline, whereas the atactic PP is completely amorphous. On the other hand, EPDM is an amorphous copolymer comprised of equal parts of ethylene and propylene with 4-5% of ethylidene norbonene to afford dynamic vulcanization. We shall first review the miscibility studies of solvent cast iPP/EPDM blends primarily based on the contributions of Kyu’s group [54, 55] and references therein. The solvent cast blend films were prepared by first dissolving iPP

48 Figure 2.9: Experimentally determined phase diagram of the PCL/PS blend by Tanaka and Nishi [53]. The solid symbols are the experimental data for the melting temperature and the open symbols are the phase separation temperatures obtained from cloud point measurements. Solid lines are ﬁts computed using the Flory diluent theory and the Flory-Huggins mixing theory.

49 powder in xylene at 130oC, then adding EPDM after lowering the temperature to

100oC and stirring thoroughly for about 90 min to assure thorough mixing. The film specimens were prepared by solvent casting at ambient temperatures in a fume hood and dried in a vacuum oven for 48 h at room temperature. The average thickness of these blend films was approximately 10-20 µm. The solvent cast films appear turbid to the naked eye, which might be attributed to the phase separation of the blends and/or the crystallization of iPP. It is difficult, if not impossible, to resolve the crystal melting and liquid-liquid phase separation especially when the melting temperature of iPP and the LCST coexistence curve of the iPP/EPDM blend are in close proximity or intersecting one another. That is to say, iPP molecules may have gained sufficient mobility during premelting, and thus it is possible that phase separation could start before the crystal melting has been completed. In order to circumvent the aforementioned problem, the cloud point measurement was performed based on light scattering by first melting the iPP crystal phase at 170oC and rapidly cooling it to 140oC within 2 min. Subsequently the sample was reheated to 190oC at a rate of 0.5oC/min. The scattered intensity was measured at an approximate

2θ angle of 20o during this reheating cycle. It was found that the blend at 140oC showed no scattering of light, suggestive of the homogenous character of the blend.

However, the intensity increased rapidly as liquid-liquid phase separation commences at about 155oC. This procedure is thermally reversible so long as the phase separation process has not advanced signiﬁcantly or the blend is not degraded. Based on this

50 Figure 2.10: Experimental phase diagram of the iPP/EPDM blend determined by Chen and Kyu [54] using light scattering and DSC techniques, showing the intersection of LCST and the crystal-melt phase transition. The solid circle symbols denote the melting temperature of the blend at various compositions obtained from heating run on the DSC trace. The solid square symbols are the DSC crystallization peaks obtained during cooling run on the DSC trace. The solid triangles are the phase separation temperatures at various compositions obtained from cloud point measurements.

51 methodology, the phase diagram for this iPP/EPDM blend was established by Chen et al. [54] as depicted in Figure 2.10. The observed phase diagram is characterized by a lower critical solution temperature (LCST) type which is intersected by the melting transition of iPP crystals. This phase diagram is reminiscent of an inverted tea-pot phase diagram of a polymer/liquid crystal system, exhibiting a liquid-liquid coexistence region, a narrow solid-liquid coexistence region, and neat crystal region bound by the solidus and liquid lines (Figure 2.10).

2.7.4 Blends of Syndiotactic Polypropylene and Ethylene Propylene Diene Monomer

To alleviate the complex phase diagram of iPP/EPDM, Ramanujam et al. [55] investigated the sPP/EPDM blend as a complementary study. The advantage of their choice of sPP is that the melting point of sPP is significantly lower than that of iPP, which could effectively decouple the mutual interference of crystallization versus liquid-liquid phase separation. Figure 2.11 displays the entire phase diagram of the solvent cast sPP/EPDM blends as established by a combination of differential scanning calorimetry (DSC) and cloud point determination. In descending order of temperature the observed phase diagram exhibits an LCST curve, followed by a crystal-liquid transition, and subsequently UCST that lies underneath it. The coexistence lines have been drawn to guide the eyes. There exists a small miscible gap between the LCST and the UCST. The thermal reversibility of the LCST can be confirmed easily. However, the interference of crystal melting makes the confirmation

52 Figure 2.11: Experimental phase diagram of the sPP/EPDM blend determined by Ramanujam and Kyu [55] using a combination of DSC and light scattering techniques, exhibiting the combined LCST (solid circles) and UCST (solid triangles) together with the melting point depression (open and solid squares). L1 +L2 represents the LCST miscibility gap. The inverted solid triangle symbols are the experimentally observed DSC peaks during cooling run. The authors suggest that S1 + L2 indicates the presence of coexistence of a solid phase with two liquid phases. The solid and open symbols at the bottom of the ﬁgure are the recorded glass transition temperatures (Tg) of the blend using DSC. Two distinct Tg can be observed suggestive of the limited miscibility of the blend. The lines are merely drawn to guide the eye.

53 of UCST more diﬃcult at least experimentally and thus it has been cautioned that the UCST should be regarded as tentative.

2.7.5 Blends of Polyethylenes of Diﬀerent Molecular Weights

Experimental phase diagrams for binary crystalline mixtures are rare. We are fortunate to notice the intriguing phase diagrams reported by Smith and Manley [47] for two polyethylene fractions of diﬀerent molecular weight, denoted 1,000 (PE1000) and 2,000 (PE2000), respectively. Thermal characterization and x-ray diﬀraction revealed the existence of a common crystalline lattice in which both constituents crystallize simultaneously. Figure 2.12 shows the single melting transitions of the two

PE fractions as obtained by the DSC experiment, showing the systematic variation of the melting temperatures. It is not surprising to discern an ideal solid solution since PE fractions contain the same PE molecules and χaa ' 0. The solidus and liquidus lines are diﬃcult to detect experimentally due to the broad nature of the experimental DSC peaks.

2.7.6 Poly(caprolactone)/Trioxane Blends

We review another interesting system investigated by Wittman and Manley, the eutectic mixture of polycaprolactone (PCL) and trioxane. PCL employed in their study had a reported molecular weight of 14,000 with a melting temperature of 334K, whereas trioxane is a high melting solvent that has a melting temperature of 335K.

Figure 2.13 shows the eutectic phase diagram of the PCL/trioxane mixture. Wittman

54 Figure 2.12: Phase diagram of a mixture of polyethylene fractions of two molecular weights 1000 and 2000 showing ideal solid solution behavior as determined by Smith and Manley [47]. The solid circles denote the experimentally observed melting temperature for various composition of the blend. Only a single melting temperature was observed suggestive of an ideal solid solution with a common melting point lying intermediate of the pure component melting temperatures. The solid line is merely to guide the eye.

55 and Manley solved these melting point depression curves in the context of the Flory diluent theory by treating one constituent at a time without the interference of the other. Their calculated liquidus lines certainly captured the trend of the melting point depression, but the solidus lines were absent by virtue of the inherent complete immiscibility assumption in the original Flory theory. There is also no description of the enthalpy of eutectic formation in the Flory diluent theory, and its applicability to eutectic mixtures is highly circumspect.

2.8 Dynamics of Crystallization of Homopolymer

In this section, we review some of the approaches taken to model the dynamics of crystallization of homopolymers. We derive the Avraami model, which is able to successfully describe the temporal dynamics of homopolymer crystallization under multiple nucleation events. Next we derive the equations of motion for the order parameter approach that has been used to get a complete spatio temporal solution to the dynamics of crystallization in homopolymers.

2.8.1 Avraami Model

Consider the model for the transition of one phase, i.e. β into another phase

α which occurs by the nucleation of α in β. Assume that, a) nuclei form randomly through out the β phase whenever the temperature is below the transition temperature, i.e., the melting temperature. Let the rate of nucleation

56 Figure 2.13: Phase diagram of a mixture of PCL trioxane mixture showing eutectic behavior as determined by Wittman and Manley [46]. The open circles are the melting temperature of the blend obtained from DSC traces. The DSC trace also has an invariant eutectic temperature peak which has is represented by the open triangle. The eutectic peak had a small unknown shoulder at a slightly higher temperature that is represented by the open squares.

57 (a) (b)

Figure 2.14: Spherulitic growth of PVDF with heterogeneous nucleation. This is an example to illustrate the growth of an α crystalline phase growing in a β amorphous phase. The PVDF was supercooled below its melting temperature by around 5oC and snapshots were taken at 1 minute intervals after a induction time of 12 mins

58 be deﬁned by an unknown function I(t) such that

dn = I(t) (2.54) dt where n is the number of nuclei generated per unit volume at time t. b) The linear growth rate is independent of the direction so that the nuclei grow as spheres until they impinge each other. Let r be the radius of the nuclei.

Z t r(t) = u(t)dt (2.55) τ

Then the linear growth rate u(t) can be deﬁned as

dr u(t) = (2.56) dt and the time at which nucleus is formed is determined as τ. Prior to impingement the volume of the nucleus formed at time τ.

4 4 Z t 3 V = πr3 = π u(t)dt (2.57) 3 3 τ

The rate of increase of the transformed volume per volume due to all nuclei which formed between τ and τ + dτ is

dV dr = 4πr2 I(τ)dτ (2.58) dt dt

59 The eﬀect of impingement can be accounted by realizing that the rate of increase in transformed volume must be proportional to the mass of the β phase remaining untransformed. Let f(t) be the volume fraction of the remaining β phase at time t.

V (t) = 1 − f(t) (2.59)

Therefore, the rate of increase of transformed volume at time t due to nuclei which formed in the interval τ and τ + dτ is

dV dr = 4πr2f(t) I(τ)dτ (2.60) dt dt

The total rate of increase of transformed volume dVT /dt at time t includes contributions from all nuclei formed between the time interval 0 to t. Thus,

dV Z t dr T = 4πf(t) r2 I(t)dτ (2.61) dt 0 dt

dV since T is the total eﬀective rate of growth of transformed volume per unit volume. dt

dV df T = − (2.62) dt dt

df Z τ=t = −4πf(t) r2u(t)I(τ)dτ (2.63) dt τ=0 df Z τ=t = −4πdt r2u(t)I(τ)dτ (2.64) f τ=0

60 Z t Z τ=t lnf = −4π r2u(t)I(τ)dτdt (2.65) 0 τ=0 Z t Z τ=t f = exp −4π r2u(t)I(τ)dτdt (2.66) 0 τ=0

The speciﬁc solution of this general equation is when we make assumptions for the unknown functions I(t) and u(t). For the case of both I(t) and u(t) assumed to be constant with respect to time, we can derive the following formula

f = exp(−kt4) (2.67)

where, πu3I k = (2.68) 3

Therefore, the transformed volume fraction is then

−kt4 VT = 1 − f(t) = 1 − e (2.69)

This is a very basic model that has been applied to determine the development of crystallinity, and the equation is referred to as Avraami equation. The only application of the theory has been in the determination of the development of crystallinity. equation predicts a sigmoidal function for the development of crystallinity as observed in experiment.

61 2.8.2 TDGL Model A

In the context of the phase ﬁeld approach to crystallization, the thermody-

R namic variable ψ has been deﬁned as anon-conserved variable i.e V ψdV is not a constant with respect to time. In order to describe the dynamics of such a order parameter we use the Ginzburg Landau Model A equations (TDGL - Model A) which can be written as ∂ψ δgc = −Γ (2.70) ∂t ψ δψ

Upon determining the functional derivative using Equation (2.43), we obtain the equation of motion,

∂ψ = −Γ [W ψ(ψ − ζ)(ψ − 1) + ∇ · κ · κ · ∇ψ] (2.71) ∂t ψ ψ ψ

where Γψ is called the mobility term and can be described as the rotational diﬀusivity of the polymer chains [35]. The free energy expression used in the TDGL-A equations can be diﬀerent as described in the previous section. The model is excellent in determining the morphological evolution of faceted single crystals as can be seen from Figure 2.15. It is also able to predict the emergence of curved single crystals or needle-like crystals. Using the appropriate form of κψ this model can explain the formation of hexagonal single crystals, square crystal or lozenge shaped crystals as seen in polyethylene. The surface free energy (σ) can take an anisotropic form as

62 Figure 2.15: From top left to bottom right we observe diamond shaped, lozenge shaped curved, lenticular curved, and slender curved single crystal that have been predicted by Mehta and Kyu [34].

63 shown by Kessler et al. [56] such that

σ = σ0β(Ω) = σ0(1 + cos(jΩ)) (2.72)

where σ0 can be taken as a reference free energy equal to σe. Ω is the orientational angle w.r.t the reference axis and j is defined as the mode such that j = 6 signifies six-fold symmetry and is used to describe the growth of hexagonal single crystals. In the next section, we will delve into some of the more striking results that are obtained by using the TDGL model A equation with the interplay of heat effects.

2.8.3 Role of Latent Heat

There is a enormous amount of latent heat generation associated with a solid-liquid transition. For an isolated system, this heat will raise the temperature of the system both locally and globally. Most real systems though are either closed or open systems. Crystallization under quiescent conditions can be considered as a problem in a closed system, unlike crystallization under flow, which would be posed as a problem in an open system. In a closed system, the thermal length scale can be used to determine the effects of thermal transport across a flat interface. In the case of polymer crystallization due to the low thermal conductivity of the polymers, the thermal length scales are usually much smaller in magnitude. This leads to the generation of stronger thermal gradients. As shown schematically in Figure 2.16, for a rough interface, the heat evolving in the valleys can get trapped causing local

64 Figure 2.16: Usually the crystalline growth front in 2D is a ﬂat crystal-melt interface. Atomic scale roughness or the presence of impurities can lead to a roughening of this interface. Under such circumstances, the heat liberated due to crystallization can get trapped locally in the valleys of the rough interface. Xu et al. [57] have shown that this can lead to intricate pattern formation such as the dendritic morphology and dense lamellar branching.

65 Figure 2.17: Xu et al. [57] have determined a morphology map using the material parameters of isotactic polystyrene. The ﬁgure shows the various possible crystalline structures that can be generated upon varying the supercooling and the extent of surface anisotropy parameter for a six fold anistropic crystal. For small undercoolings (bottom two rows), we observe the formation of a spherulite growth with very little or no internal structure (far left) and faceted single crystal (far right). For larger under coolings (Top two rows), we observe spherulite growth with seaweed and dense lamellar branching morphologies on the far left and the formation of dendrite morphology on the far right.

66 melting. The mechanism of dendritic growth has long been attributed to this eﬀect.

For dendritic growth a strong anisotropy is required in the surface free energy i.e is high. In order to incorporate the heat effects, we solve an additional heat diffusion equation concurrently with the crystallization field.

∂T ∂ψ ρC = ∇k ∇x + ∆H (2.73) p ∂t T u ∂t

where Cp is the speciﬁc heat of the polymer, ρ is the density, kT is the thermal conductivity and ∆Hu is the latent heat of crystallization. The last term is the heat source term to account for latent heat generation at the interface. Recently, we have shown that supercooling and strength of anisotropy play a vital role in determining the ﬁnal morphology of a material. From the morphology landscape of simulated results of polystyrene, we can observe that the single crystals are obtained at lower

∆T and higher strength of anisotropy. On the other hand spherulitic morphology or the dense lamellar branching morphology is observed for a much higher ∆T but much weaker strength of anisotropy leading to the conclusion that there is deﬁnitely some relation between the two. Lack of experimental techniques for the determination of the surface free energy prevent further elaboration of this idea.

2.9 Dynamics of Liquid-Liquid Demixing in Binary Mixtures

The dynamics of phase separation can be characterized by two diﬀerent mechanisms. The location of the system on the phase diagram according to absolute tem-

67 perature and composition generally determines the mechanism of phase separation as seen in Figure 2.2. In the metastable region of the phase map nucleation and growth

(NG) usually occurs if there is a large enough concentration ﬂuctuation to trigger the process. In the unstable or spinodal region, the system will most likely phase separate by means of a spontaneous decomposition process termed as spinodal decomposition

(SD) [58, 59]. Although the equilibrium structures are not always discernible from one another, the initial and intermediate structures during the course of the phase separation are quite different for the two mechanisms. In the early stages of SD, periodic concentration fluctuations with a particular wavelength are built up throughout the entirety of the sample space and an amplitude of fluctuation, ∆φ , increases with time, while the wavelength of fluctuations remains essentially constant [5]. The wavelength is influenced by the thermodynamic conditions of the mixture characterized by the quench depth while the amplitude of the fluctuations are determined by the kinetics, or alternatively speaking, the time of the phase separation.

2.9.1 Nucleation and Growth (NG)

In the metastable regions of the phase diagram, the continuous phase gives birth to intermediary unstable embryos due to excesses in free energy. These embryos later play the role of nuclei possessing the equilibrium composition, subsequently growing by a mechanism of downhill diﬀusion. During the course of the growth of the nuclei, the bigger nuclei grow at the expense of the smaller ones. This is a characteristic of the nucleation and growth controlled mechanism wherein the largest

68 (a) (b)

Figure 2.18: Images on the left show the progression of binodal phase separation and the images on the right are their Fourier transforms which can be used to determine the size of the phase separated domains.

69 fluctuations in magnitude and size usually survive the longest. This phenomenon is fundamentally different from coalescence as no actual convective flows exist, driven completely by diffusion, it is known as Ostwald ripening [29]. A nucleus possesses an excess surface energy that induces an aggregation process forming a new stable phase. This activation energy, ∆G∗, required to initiate the NG process is expressed as 4 ∆G∗ = − πr3∆G + 4πr2σ (2.74) 3 v where the first term is the free energy contribution of the volume of the nucleus and the second term is the energy required to form an interface between the nucleus and the mother phase. ∆Gv is the free energy difference between a nucleus and the mother phase while σ is the interfacial energy per unit area and r is the radius of the nucleus that can be calculated by minimization of the activation energy with respect to r to yield the relation, rmin = 2σ/∆Gv. When nuclei form, the system decomposes with a decrease in the free energy causing nucleus growth resulting in droplet domains.

2.9.2 Spinodal Decomposition (SD)

Cahn and Hilliard [5] were the ﬁrst to discus the spontaneous phase separation of mixtures via spinodal decomposition in binary alloys of metals. This particular process of phase separation does not require an activation energy unlike NG mechanism, but proceeds spontaneously in the presence of minimal concentration

ﬂuctuations or thermal noise. SD can be classiﬁed by three stages according to time:

70 (a) (b)

Figure 2.19: Images on the left show the progression of spinodal phase separation and the images on the right are their Fourier transforms, which can be used to determine the size of the phase separated domains.

71 early, intermediate, and late. The early stage can be characterized by a linearized diﬀusion equation which has been solved analytically. There is, however, no analytical solution to the intermediate and late stages of SD, and a scaling law is generally applied to describe these regimes. Phase separation is a diﬀusion process and can be described by a continuity equation [5]

∂φ = −∇J(r, t) + η(r, t) (2.75) ∂t where J(r, t) is the diffusion current dependent on space and time. This diffusion current is proportional to the gradient of the local chemical potential difference, µ(r).

J(r, t) = −M∇µ(r) (2.76)

where M is the mobility related to the Onsager coeﬃcient [60] (Λ) and Λ = M/kBT .

For a two-component system, Λ obeys the Onsager reciprocity,

Λ Λ Λ = 1 2 (2.77) Λ1 + Λ2

where Λ1 and Λ2 are self-diffusion coefficients of the two components. η(r, t) represents the random force due to the Brownian motion known as thermal noise. The strength of this thermal noise is directly related to the mobility M according to the fluctuation- dissipation theorem. Cahn and Hilliard [5] first formulated the mathematics for

72 the spinodal decomposition in metal alloys and polymer mixtures by employing the

Ginzburg-Landau free energy expansion to an inhomogeneous mixture,

∆G Z gi = m = f(φ, r) + κ(∇φ(r))2 dV (2.78) kBT V where gi, f(φ, r), φ(r), and κ are the total free energy density, local free energy density, local concentration, and concentration gradient coefficient, respectively. The gradient term in the free energy quantifies the energy required to form interfaces in inhomogeneous mixtures. de Gennes [60] modified the Cahn-Hilliard theory by replacing the local part of the free energy density by the Flory-Huggins free energy density and relating the gradient term coefficient κ to the Kuhn segment length, a, of symmetric polymer blends by the random phase approximation

a2 κ = (2.79) 36φ(1 − φ)

Binder further modiﬁed this equation for asymmetric binary polymer blends.

1 a2 a2 κ = 1 + 2 (2.80) 36 φ 1 − φ

73 The local chemical potential µ can be evaluated by taking the functional derivative of the total free energy of the system with respect to composition φ.

δgi ∂gi ∂gi µ = = − ∇ (2.81) δφ ∂φ ∂∇φ

Substitution of Equation (2.81) back into the diﬀusion equation yields the time- dependent Ginzburg-Landau Model B equation for a conserved concentration order parameter φ. ∂φ δgi = M∇2 + η(r, t) (2.82) ∂t δφ

After calculating the expression for the functional derivative, the continuity equation for an arbitrary asymmetric binary system becomes

∂φ 1 + ln(φ) 1 + ln(1 − φ) = M∇2 − + χ(1 − 2φ) + 2κ∇2φ (2.83) ∂t X1 X2

This form of the equation maintains a constant mobility term and noise is neglected since it does not play a signiﬁcant role in spinodal decomposition [60] though it can be added as desired. κ can be assumed to be a scalar constant since the concentration dependence is required to accurately model only the late stages of phase separation [61]. Also the square of the concentration gradient term can be ignored in the case of slow spatial variation since it is quite small relative to other contributions.

This equation is also referred to as the TDGL-B equation for conserved order parameter. The analytical solution of this equation exists for the spinodal region, which is

74 critically unstable. By solving the linearized diﬀused equation in fourier space in 2 dimensions, Cahn and Hillard [5,58,62–64] found that the emerging morphology was a superposition of sine waves that were randomly distributed in space both in direction and phase. Subsequent computer calculations conﬁrmed this result wherein the superposition of around 20 sine waves was enough to recreate the spinodal structures seen in experiment.

2.10 Dynamics of Crystallization in Polymer Blends

In this section, we review some of the experiments and simulations to investigate the concentration profiles and morphologies that occur during crystallization in blends. The experiments of Billingham et al. [65], conducted with isotactic polypropylene(iPP) mixed with atactic propylene(a-PP), serve as a good starting point since only one component is crystallizable in this blend and the components are presumably homogenous and well-mixed in their melt states. The theoretical analysis of the dynamics of solid-liquid phase separation is limited and the work by Mcfadden et al. [?] on the development of concentration profiles during solidification of ideal metal alloys is one of the few simulations that capture the transient and steady state concentration profiles during solidification in binary crystalline systems.

Billingham et al. [65] studied the rejection of non-crystalline materials from a crystallizing front in a blend of isotactic polypropylene (iPP) and atactic polypropylene (aPP). The uniqueness of their approach is that aPP was labeled with a ﬂu-

75 Figure 2.20: Snapshot of a iPP spherulite growing in a 90/10 iPP-aPP blend superimposed with the concentration profiles of the aPP component along the center line. This experiment was conducted by Billingham et al. [65] using UV fluorescence spectroscopy. The a-PP fraction was modified with a marker molecule that is UV fluoroscence active to monitor the concentration of the atactic fraction within and outside the spherulites of iPP.

76 orescent probe enabling the rejection of labeled aPP from the emerging spherulitic morphology to be observed directly under a fluorescence microscope. Figure 2.20 shows the snapshot a growing spherulite of iPP in 90/10 blend of iPP-aPP. A line scan of the intensity of the UV fluorescence through the centerline of this spherulite gives us the concentration profile of the a-PP component. We can see that there is rejection of the a-PP at the interfaces exhibited by a sharp increase in the a-PP concentration at the interfaces. There is also a significant amount of aPP that is left within the spherulite evidenced by the rather small drop in a-PP concentration inside the spherulite.

McFadden et al. [?] used their free energy forms developed using order parameters, and used the TDGL Model A and B equations to determine the concentration profiles in ideal solid solutions of Cu-Ni alloys. Their results are very similar to experimental results of Billingham showing a rejection front and virtually no difference between the concentration of the solid phase and the liquid phase. It is tempting to assume that this explains the results of Billingham. On the contrary, the simulations of McFadden et al. [?] are for a ideal solid solution of Cu-Ni where the difference in the solidus and liquidus concentration is almost negligible. Hence, they observed no differences in concentration between the solid and liquid phases. For the case of the iPP-aPP blend which in principle should crystallize out pure iPP, the concentration of the solidus phase should be 100% iPP. This leads us to surmise that there must be some interplay of thermodynamics and kinetics which result in the emergence of the

77 concentration proﬁles seen in iPP-aPP blends. This phenomenon is very intriguing and will be investigated in detail in the subsequent chapters.

78 CHAPTER III

BLENDS OF CRYSTALLINE AND AMORPHOUS POLYMERS

3.1 Introduction

Over a half century ago, solubility of a polymer solute in a monomeric solvent was theoretically deduced by Flory by relaxing two assumptions inherent in the regular solution theory; viz. [24] (i) the constituent polymer solute and solvent are dis- similar and (ii) the mixture is non-ideal, except that Flory kept the last assumption:

(iii) the complete insolubility of solvent in the pure solid crystal. This regular solution theory, known as the Flory diluent theory, has been traditionally used to describe the melting point depression of crystalline polymer solutions [66]. The Flory diluent theory for polymer solutions was later extended by Nishi and Wang [26] to determine the Flory-Huggins interaction parameter, χ, from the melting point depression data of PVDF and PMMA polymer blends under the aforementioned assumption of complete immiscibility [6] of the polymeric solvent (PMMA) in the pure PVDF crystal.

Since then, the ﬁeld of polymer blends has enjoyed explosive growth by virtue of the ease of the melting point experiment and the simplicity aﬀorded by the analytical expression of the Flory diluent theory. The χ value as obtained by this melting point depression approach is generally larger (e.g., order of magnitude) relative to those of

79 small angle neutron and x-ray scattering [52]. Although such a discrepancy in the

χ values between the above approaches has been noticed, little attention has been paid to the true physical meaning of the χ interaction parameter used in the melting point depression approach. It should be emphasized that most neutron scattering experiments are generally done in the melt, so it may be attributed unambiguously to the segmental amorphous–amorphous interaction as deﬁned in the original Flory

Huggins theory for amorphous-amorphous polymer blends. However, the meaning of the χ interaction parameter becomes obscured in the melting point depression analysis because it was not clearly deﬁned whether the χ interaction parameter as obtained from the melting point depression represents the amorphous-amorphous interaction, crystalline chain-amorphous interaction, or the combination of both. This ambiguity went unchecked despite the reported discrepancy between the χ interaction parameter values obtained by the melting point depression and by other techniques such as neutron scattering experiments.

Moreover, the thermodynamic phase diagrams of polymer blends containing one or both crystalline component(s) are at odds with other disciplines including small molecule organics, metal alloys, and liquid crystal mixtures where the solidus and liquidus lines are well-deﬁned forming eutectic and peritectic phase diagrams [67].

Guenet et al. [68–70] have experimentally determined phase diagrams of syndiotactic polystyrene with several solvents where the crystalline phase of the blend is not pure.

It can be reasoned that the Flory theory must be valid for completely immiscible

80 crystalline-amorphous systems and captures the liquidus line, but it is incapable of describing the solidus line due to the explicit assumption of the complete immiscibility of the solvent in the solute crystal. A natural question is what happens to the phase diagrams if this complete insolubility assumption is removed.

In this chapter, we relax the last assumption of complete immiscibility of the solvent in the crystal by incorporating crystalline-amorphous interaction parameter in addition to the amorphous-amorphous interaction parameter. A variety of phase diagrams for a crystalline-amorphous polymer blend has been constructed by incorporating the amorphous-amorphous and crystalline-amorphous interactions of the constituent polymers in the free energy description. Various coexistence curves have been solved through global free energy minimization in conjunction with a common tangent method. The rigor of the present theory has been tested in comparison with the reported experimental observations.

3.2 Free Energy Landscape

We shall consider a polymer blend in which only one component can crystallize and the other is a non-crystallizable amorphous polymer. The total free energy density of mixing of a crystalline-amorphous polymer blend may be expressed as the weighted sum of the free energy density pertaining to crystal order parameter [57] of the crystalline constituent with its volume fraction (φ) and the free energy of liquid-liquid mixing as described by the Flory-Huggin’s theory of mixing [1]. The

81 free energy density of solidiﬁcation was weighted by its volume fraction to ensure that the solidiﬁcation potential vanishes in the limit of zero concentration of the crystalline constituent. As pointed out earlier, we relax the last assumption retained in the Flory diluent theory by taking into consideration amorphous-amorphous and crystalline-amorphous interactions, viz.,

φ (1 − φ) 2 f(ψ, φ) = φf(ψ) + ln(φ) + ln(1 − φ) + {χaa + χcaψ }φ(1 − φ) (3.1) r1 r2

where χaa is the Flory-Huggins interaction parameter representing the amorphous- amorphous interaction of the constituent chains in the isotropic melt. χca represents the crystalline-amorphous interaction parameter. Note that the order of the subscripts denotes the constituent 1 (crystal) and constituent 2 (amorphous polymer). This nomenclature is followed throughout the course of the dissertation. In the present model, the crystallizable constituent is treated as component 1. Its concentration φ1 is denoted as φ by dropping the subscript. The concentration of the amorphous component is φ2 and is equal to 1 − φ1 ≡ 1 − φ. This also follows for

the amorphous-amorphous interaction parameter (χaa ≡ χa1a2 ) and the crystalline-

amorphous interaction parameter (χca ≡ χc1a2 ). For this chapter, this nomenclature is also applied to the crystal order parameter of component 1 (ψ1) and it is denoted by ψ. In the next chapter, we will extend this model to binary crystalline blends where the crystal order parameters for each component are written explicitly with appropriate subscripts. r1 and r2 correspond to the statistical segmental lengths of

82 the respective components. Higher order couplings can also exist between the φ and

ψ order parameter though the strength of such interactions is usually weaker. We have used the lowest order coupling that recovers the free energy of the crystallization of the homopolymer (f(ψ, φ) → f(ψ) as φ → 1). Also, f(ψ, φ) → 0 as φ → 0 and minimization of ∂f/∂ψ retains a zero root that signiﬁes the melt is unaﬀected by crystalline-amorphous interactions. That being stated, although the explicit entropy change upon crystallization could be accounted by using a weighting factor for the entropy terms of the Flory Huggin theory of mixing, this free energy description takes into consideration both the enthalpy and entropy change using the form described in

Equation (3.1).

To clarify the physical essence of the crystalline-amorphous interaction, Equa- tion (3.1) may be rewritten as

φ (1 − φ) f(ψ, φ) = φf(ψ)+ ln(φ)+ ln(1−φ)+χaaφ(1−φ)+χca[φψ][(1−φ)ψ] (3.2) r1 r2 where the crystal phase order parameter ψ can be deﬁned as the ratio of the lamellar thickness l to the lamellar thickness of a perfect crystal l0, i.e., ψ = l/l0, and thus it represents the linear crystallinity (i.e., one-dimensional crystallinity) of the crystallizing component. Then the product of φ and ψ in the last term of Equa- tion (3.2) roughly corresponds to the bulk crystallinity in the blend, whereas the product of (1 − φ) and ψ implies the amount of amorphous materials interacting with the crystalline phase, and hence the last term, χcaφψ(1 − φ)ψ, signiﬁes the

83 crystalline-amorphous interaction. In order to estimate the crystalline-amorphous interaction parameter we use the following arguments. The crystalline-amorphous interaction is a measure of the energy required for a solvent to exist in the crystalline phase. Guenet et al. [68, 69] have reported experimental phase diagrams for blends of syndiotactic polystyrene with various solvents (naphthalene, benzene, and toluene) that show the existence of the solvents within crystalline phases and are termed crystallo-solvates or intercalates. In the Flory diluent theory, the energy of mixing of crystalline and amorphous constituents is assumed to be very large (inﬁ- nite) thereby causing the crystalline component to crystallize into the pure component from a crystalline-amorphous melt blend. In the description of the thermodynamics of ordering of liquid crystalline molecules and liquid crystalline polymers, the mixing energy of the ordered and disordered components is equivalent to the enthalpy of ordering of the liquid crystalline component [41]. Thus, we propose that a crystalline- amorphous interaction energy exists. The crystalline-amorphous interaction parame-

ter is a scaled form of the interaction energy (χc1a2 = ∆Hc1a2 /RT ). This is analogous to the amorphous-amorphous interaction energy (∆Hm) which is normalized with the thermal energy scale (RT = NAkBT ) to yield the amorphous-amorphous interaction energy parameter (χaa). In the course of the dissertation, we will report the values of the amorphous-amorphous and crystalline-amorphous parameters employed accompanied by a reference temperature (Tref ). The actual crystalline-amorphous interaction energy can easily be back-calculated using the relation χcaRTref . The

84 determination of the rest of the model parameters is outlined in Appendices B and

C. In the next section, we investigate the role of χca and χaa in the construction of phase diagrams.

The aforementioned crystal order parameter (ψ) may be described in terms of the Landau-type free energy expansion, viz.

F (ψ) ζ(T )ζ (T ) ζ(T ) + ζ (T ) 1 f(ψ) = = W 0 m ψ2 − 0 m ψ3 + ψ4 (3.3) kBT 2 3 4 where the coefficients of the Landau free energy expansion are treated as temperature dependent so that the free energy has the form of an asymmetric double well at a given crystallization temperature or supercooling, but it reverts to the symmetric double well at equilibrium. This kind of asymmetric Landau potential has been utilized in the phase field model to explain the solidification phenomena such as crystallization

[57]. It should be cautioned that the coefficient of the cubic power term must be non-zero in order to apply the Landau potential to the first-order phase transition; otherwise, Equation (3.3) is applicable only to a second-order phase transition or to equilibrium. ζ represents the unstable hump for the crystal nucleation to overcome the energy barrier and W is the coefficient that represents the penalty for the nucleation process. ζ0 represents the crystal order parameter at the solidification potential of crystallization that is treated to be crystal melting temperature dependent [57].

The uniqueness of the present free energy description is that the crystal order parameter at the solidiﬁcation potential strongly depends on the supercooling,

85 and thus its value is less than unity at a given crystallization temperature implying that the crystallinity is not perfect. In other words, the emerged polymer crystal is metastable (i.e., non-equilibrium) and thus imperfect, reflecting the polycrystalline nature of polymer crystals that rarely reach any thermodynamic equilibrium. This solidification potential shifts with supercooling, and thus the size and shape of the structure as well as the degree of crystallinity would be different for different supercooling. This type of asymmetric Landau potential has been employed successfully to elucidate the dynamics of crystallization and morphology evolution of neat syndiotactic polystyrene showing the spatial and temporal emergence of hierarchy structures encompassing faceted hexagonal single crystals, snowflake, seaweed, and dense lamellar morphology (i.e., spherulite). In the present chapter, we shall focus only on equilibrium aspects of thermodynamic phase diagrams of crystalline-amorphous polymer blends.

3.3 Completely Miscible Systems

Figure 3.1(a) shows the free energy landscape of a hypothetical crystal/amorphous blend in which free energy exhibits dual minima with respect to the crystal order parameter (ψ) and a single minimum with respect to the compositional order parameter

(φ). Phase diagrams were established via global minimization of the free energy density by ﬁrst minimizing the free energy density, f(φ, ψ), with respect to the crystal order parameter (ψ) based on the steepest descent algorithm with a tolerance of 10−7

86 Figure 3.1: Free energy landscape representing (a) a hypothetical miscible binary mixture showing a single minimum on the front surface (i.e. φ at ψ = 0,) and the dual minima on the side surface on the right (i.e., ψ at φ = 1) and (b) a hypothetical partially miscible binary mixture exhibiting dual minima on both the front surface (i.e., φ at ψ = 0) and the side surface on the right (i.e.,ψ at φ = 1). The solid dots indicate the roots of ∂f/∂ψ = 0. The thermodynamic parameters used are W = 10, ζ = 0.1, ζ0 = 1, r1 = 1 and r2 = 1. a) χaa = 0 b) χaa = 3.

87 in determining the roots at ∂f/∂ψ = 0 (Figure 3.1(a) - solid dots). Subsequently, the pseudo-chemical potentials (∂f/∂φ) are calculated by taking derivatives with respect to volume fraction (φ) and the coexistence loci were determined with the aid of the common tangent algorithm [71]. It has to be noted that the global minimization of f(φ, ψ) can be carried out by the tangent plane minimization also which is numeri- cally cumbersome. We have used the fact that the crystal order parameter (psi) is a non-conserved quantity. This implies that at the global free energy minimum, the free energy is minimum with respect to ψ at the coexistence phase concentrations.

For illustration purposes, a hypothetical polymer phase diagram was ﬁrst calculated showing an overlap of the solidus and liquidus lines (Figure 3.2(a)), assuming heat of crystallization to be 3 kcal/mole, r1 = 10 0, r2 = 3 0 and χca = 0.01 at the melting temperature (500K) of the neat crystallizable component. χaa was taken negative by setting a high temperature LCST located at 600K. We used the relation

χaa = A + B/T = A + (χcrit − A)Tcrit/T to determine χaa. We calculate χcrit = 0.4 and set A = 3 and B = −1560 to get χaa = −0.12 (see Appendix B). With increasing repulsive crystalline-amorphous interaction parameter to χca = 0.1 , the solidus and liquidus lines get separated showing the coexistence of the crystal + liquid (amorphous) region (Figures 3.2(b)). As χca increases, the solidus and liquidus curves both shift downwards, but the crystalline-amorphous coexistence gap broadens, implying that the solidus line moves down much faster than the liquidus curve.

88 Figure 3.2: Effect of increasing crystalline-amorphous interaction parameter χca= (a) 0.01 (b) 0.1 (c) 0.3 at 500K. As χca increases the solidus and liquidus lines move downward and the crystal-liquid coexistence region broadens till the solidus line hits the pure component axis. Figure (d) shows the effect of increasing the amorphous-amorphous interaction energy for a finite value of crystalline- amorphous interaction parameter χca = 0.3 at 500K. The liquidus curve shifts downward as the intermolecular interaction of the liquid becomes more attractive. In (a), (b) and (c), χaa was calculated by setting A=3 and TLCST = 600K. In (d) χaa was progressively lowered by changing A=3, 4 and 5 and TLCST = 600K.

89 At the crystalline-amorphous interaction parameter of χca = 0.3 , the gap between the solidus and liquidus lines becomes very wide such that the solidus line is located very close to or right onto the pure crystal ordinate, while the liquidus line shows appreciable depression with increasing amorphous constituent (Figure

3.2(c)). This trend remains virtually unchanged with further increase of the repulsive crystalline-amorphous interaction parameter, provided that χaa is kept constant.

This in turn suggests that the χca, which is more sensitive to the solidus line, reaches its limiting value for complete rejection of the solvent molecules from the crystallizing front. This is exactly what Flory pointed out in his diluent theory to justify the ﬁnal assumption of complete immiscibility between the crystalline-amorphous blends. The actual crystalline-amorphous interaction energy for χca = 0.3 is equal

th to χcaRTref = 0.3 ∗ R ∗ 500 = 297cal/mol. This is about 1/10 of the enthalpy of crystallization used for the calculation of the hypothetical phase diagrams.

Figure 3.2(d) demonstrates the eﬀect of attractive amorphous-amorphous interaction parameter (i.e., negative χaa) on the liquidus line at the limiting value of crystalline-amorphous interaction parameter of χca = 0.3 . The liquidus curve shifts downward if the intermolecular liquid-liquid interaction becomes more attractive; otherwise, it moves up if the liquid-liquid interaction becomes less attractive. It may be hypothesized that the Flory diluent theory provides only an upper bound to the determination of χaa. Only in the case of completely miscible crystalline-amorphous blends, it might be possible to determine the χ interaction parameter from the melting

90 point depression that represents the χaa amorphous-amorphous interaction at a given crystalline-amorphous interaction value of χca, which must be speciﬁed.

To substantiate the above hypothesis, the calculated solidus and liquidus lines are depicted in Figure 3.3 in comparison with the melting point depression data of the crystalline-amorphous blend of poly(vinylidenediﬂuoride) (PVDF) and poly(methylmethacrylate) (PMMA) reported by Nishi and Wang [26]. In the calculation, the statistical segment values of r1 = 3371 and r2 = 866, as estimated from the respective molecular weights of PVDF and PMMA, were utilized and heat of crystallization of 1.6 kcal/mol. From the theoretical ﬁt with the experimental melt-

o ing points, we obtained a value of χaa = −0.141 at 165 C (based on the χca value of

0.58 for the above mentioned r1 = 3371 and r2 = 866). It should be cautioned that the present χca parameter is molecular weight dependent as in the case of the FH interaction parameter, χaa. We back-calculate the crystalline-amorphous interaction energy equal to χcaRTref ' 0.5 kcal/mole. χca can be higher also since the reported value of χca was the lowest value at which the liquidus line stopped changing and the solidus line lies on the pure component axis. Further change in the liquidus line can only be brought by changing χaa (or A). In order to ascertain that the blend is miscible near the melting temperature, the above calculations used the LCST temperature of the PVDF/PMMA blend to be 330oC [72]. Correspondingly, to calculate

χaa (see Appendix B), we set A = 0.37 and B = −227.8. We used an estimated value of χaa = −0.141, smaller in magnitude (i.e., lesser attractive interaction) than

91 Figure 3.3: Theoretically calculated phase diagram of PVDF/PMMA blend (indicated by ﬁlled dots) and experimental data shown by open diamonds (♦). The experimental data was determined by Nishi and Wang [26]. The calculated liquidus line coincides with the experimentally obtained melting point data. The solidus line is very close to the pure component axis. We denote the coexistence region between the solidus and liquidus lines as Cr+L. The blend exists in the isotropic melt state above the liquidus line. An LCST temperature of 330oC was used in the present calculation. (LCST curve outside scale)

92 the reported result by Nishi and Wang, i.e., χmp = −0.31 obtained from the melting point depression results in the vicinity of the melting temperature of pure PVDF, but it is much closer to that of the small-angle neutron scattering data of the same

PVDF/PMMA blends, e.g., χaa ∼ −0.226 and -0.077 for two diﬀerent concentrations of PVDF/PMMA by Stein and coworkers [73] and -0.14 by Canalda et al. [52], and

-0.03/-0.16 from SAXS by Wendorﬀ [74]. In the Flory diluent theory, the repulsive crystalline-amorphous interaction χca was completely ignored, and thus it is possible that the χaa had been overestimated as compared to those of other experiments. If the χaa value were determined directly from an independent experiment such as SANS or SAXS in the isotropic state, the actual value of χca could be obtained from the melting point depression data. Though this is one possible route to determine χca, it is a daunting task to know the complete behavior of χaa as a function of temperature and concentration. For the case of the PVDF/PMMA system, we have already assumed the A parameter because of the non-ideal behavior of the melt blends that

o shows an LCST at TLCST = 330 C. In order to get the right estimate of A we would need to know the actual experimentally observed LCST coexistence curve which was not available for this particular blend. This provides motivation to conduct the experiment to construct the experimental phase diagram which can be pursued in the future. It is surprising to note that even in the present calculation, the phase diagram could not be calculated without using a negative amorphous-amorphous interaction parameter. This kind of uncertainty may be eliminated completely in a partially mis-

93 cible system where the solid-liquid phase transition is competing with the UCST type liquid-liquid phase separation. For a UCST system that follows typical behavior we can show that the χaa can be calculated from the experimentally observable TUCST and a knowledge of r1 and r2 alone.

3.4 Partially Miscible Systems:

In the case of non-ideal mixing, liquid-liquid phase separation is expected to occur in competition with the melting transition of the crystallizable component [75].

Figure 3.4 shows a series of predicted phase diagrams for mixtures exhibiting both solidus and liquidus lines experiencing the inﬂuence of the upper critical solution temperature (UCST) type liquid-liquid demixing. As depicted in Figure 4a, the UCST peak is located at a lower temperature (450K) below the melting transition (500K) of the crystalline constituent. With increasing contribution of the liquid-liquid demixing, the UCST (Tc = 550K) or the χaa amorphous-amorphous interaction parameter, the binodal curve tends to protrude through the liquidus line, showing a teapot type phase diagram having various coexistence regions encompassing isotropic, liquid- liquid, solid-liquid, and pure solid regions. This kind of phase diagram has been commonly observed in metal alloys, organic liquids, and anisotropic liquid crystal mixtures. The major advantage of the present model over the existing theories such as the Flory diluent theory is the capability of predicting the solidus line that has been overlooked in the determination of polymer phase diagrams due to the poor as-

94 Figure 3.4: Predicted phase diagrams for crystalline-amorphous mixtures exhibiting (a) isotropic, crystal-liquid, neat crystal region, and the melting transition of the crystallizable component and (b) a tea-pot type phase diagram showing liquid-liquid coexistence region as the upper critical solution temperature envelope protruded above the liquidus line of (a). The L+L region denotes the region of liquid-liquid demixing and the Cr+L region denotes the region of solid-liquid demixing.

95 sumption of the complete insolubility of the solvent in the polymer solute. In practice, the solidus line is hard to obtain experimentally due to the non-equilibrium nature of polymer crystallization and the uncertainty of the broad DSC peaks [47]. Occasion- ally, even if two distinct DSC peaks were observed experimentally, the interpretation has been biased towards the polydisperse nature of the crystalline polymer leading to the broad or multiple peaks. Another possibility is the lack of proper understanding of the Flory diluent theory, especially the complete immiscibility assumption, and thus only a single line has been drawn in literature to represent the crystal-melt transition in the binary crystalline-amorphous blend. In certain situations, the solidus and liquidus lines may be too close to be diﬀerentiated experimentally.

A typical example is the phase diagram of the blend of polycaprolactone

(PCL)/low molecular weight polystyrene (PS) in which a single melting transition of

PCL was intersected with the UCST type liquid-liquid coexistence curve [53]. The experimental data were replotted in Figure 3.5(a) to compare with our calculated phase diagram using the statistical chain lengths, i.e., r1 = 1 and r2 = 37 that correspond to the weight-molecular weights of the respective constituents, i.e., 950 for

PS and 33000 for PCL. The UCST peak is located at 510K higher than the melting temperatures of PCL (333K) in the blends. χaa was estimated directly from the

UCST, i.e., χaa = 1.04 at 333K. Using this χaa value, the melting point depression curve was constructed according to the original Flory diluent theory, which ended predicting a much suppressed liquidus trend (the crosses in Figure 3.5(b)), indicating

96 the failure of the Flory diluent theory for this partially miscible PCL/PS blend. As advocated by the present theory, the crystalline-amorphous interaction parameter, χca and the amorphous-amorphous interaction parameter, χaa, contribute to the UCST as well as the melting transition behavior. Since χaa is already known from the UCST part and the heat of crystallization of PCL is 3.69 kcal/mol, the repulsive crystalline- amorphous interaction parameter is evaluated to be χca = 0.46 at 333K from the solidus line. Evidently both solidus and liquidus curves, although very close, are certainly consistent with the experiment. Thus the present approach demonstrated that the amorphous-amorphous interaction parameter, χca can be calculated directly from the UCST part of the phase diagram, whereas the χca crystalline-amorphous interaction parameter can be determined from the crystal-melting transition.

3.5 Conclusions

We have developed a new thermodynamic model for the determination of phase diagrams of a crystalline polymer solution. The modiﬁcation was made to the original Flory diluent theory; i.e., we relax the complete immiscibility assumption of the polymeric solvent in the neat solid crystal by taking into consideration the crystalline-amorphous interaction in addition to the amorphous-amorphous interaction of the pair. Ignoring the crystalline-amorphous interaction χca, the original

Flory diluent theory may result in over-predicting the χaa value. There has been limited success in the calculation of the phase diagrams of PVDF/PMMA blends

97 due to the absence of the experimentally determined LCST coexistence curve. For a partially miscible system, the present theory permits the simultaneous determination of χca and χaa from the melting transition and the UCST envelope, respectively.

The calculated phase diagrams of crystalline-amorphous polymer blends capture both solidus and liquidus lines forming monotectic phase diagrams consistent with those of other systems such as metal alloys, organic crystals, and liquid crystals. In addition, the original Flory diluent theory is recovered at the limit of complete immiscibility assumption of solvent and solute crystals.

98 Figure 3.5: Comparison between the experimental phase diagram of PCL/PS blend by Tanaka and Nishi (denoted by open diamonds (♦)) and the calculation represented by the solid line. (b) The enlarged melting transition region of the phase diagram showing the ﬁt by the present theory (the ﬁlled solid lines) as opposed to the melting point depression (denoted by the crosses (x)) calculated by the original Flory diluent theory using the value of χaa = 1.04.

99 CHAPTER IV

BINARY CRYSTALLINE BLENDS

4.1 Introduction

Eutectic crystallization has been extensively explored in metal alloys and small molecule systems [36, 76] but such a phenomenon has received little attention in binary crystalline polymer blends [67], perhaps due to its rare occurrence in polymer crystallization. In general, the eutectic crystal is characterized by a eutectic point in the phase diagram where two solid phases are in simultaneous equilibrium with the isotropic liquid phase. Smith and Pennings were the ﬁrst to investigate the eutectic crystallization of isotactic polypropylene and pentaerythrityl tetrabromide [77]. Sub- sequently, Manley et al. reported the eutectic crystallization of poly(caprolactone) in trioxane as well as in blends of polyethylene fractions [45,47]. The observed eutectic curves were analyzed in the framework of the Flory diluent theory [7] by individually treating the melting point depression of the constituent crystals as independent of each other. Although their calculation was found to correlate well with the liquidus line of one of the constituents, the solidus lines are absent. This is one of the major deﬁciencies in the original Flory diluent theory which is incapable of explaining the solidus line in the phase diagram because of the inherent immiscibility assumption

100 between the amorphous liquid and the neat solid crystal, that is to say, the chemical potential of the liquid solution was equated in the derivation to that of pure polymer crystal. The goal of the present chapter is to elucidate the phenomenon of eutectic, peritectic or azeotropic crystallization in polymer blends by taking into consideration the solid solution phase which has been known to exist in other binary systems such as metal alloys, organic molecular solutions, and liquid crystalline mixtures. A theoretical model has been deducted in the framework of the phase field model of solidification involving Landau-type double-well potential pertaining to the first order solid-liquid phase transition [29, 34, 35, 43, 57, 78] coupled with the Flory-Huggins free energy for liquid-liquid demixing [3]. To demonstrate the predictive capability, various eutectic, peritectic, or azeotrope phase diagrams of polymer mixtures have been established as a function of anisotropic interaction parameter and also of the amorphous-amorphous interaction parameter. The calculated eutectic phase diagrams exhibit two solidus lines and two liquidus lines corresponding to the individual crystallizing components that are in equilibrium at the eutectic point, covering solid-liquid, liquid-solid, and solid-solid coexistence regions. The validity of the proposed free energy description of the polymer solidification has been tested favorably with the experimental phase diagrams reported in literature.

101 4.2 Phase Field Free Energy of Crystallization of a Homopolymer

The free energy density of crystal solidiﬁcation pertaining to the crystal phase order parameter (ψ) may be described in the context of the Landau-type asymmetric potential (Figure 4.1), viz.

F (ψ) Z ψ f(ψ) = = W ψ(ψ − ζ)(ψ − ζ0)dψ (4.1) kBT 0 where the coefficients of the Landau free energy expansion in terms of a crystal order parameter (ψ) are treated as temperature-dependent in polymer crystallization to account for the imperfect nature of polymer crystals. This crystal phase order parameter may be defined as the ratio of the lamellar thickness (l) to the lamellar thickness of a perfect polymer crystal (l0), i.e., ψ0 = l/l0, and thus it represents the one-dimensional crystallinity, hereafter called linear crystallinity [34, 57]. This kind of asymmetric Landau potential has been utilized in the phase field model to explain the dynamics of solidification phenomena such as crystal growth. It should be cautioned that the coefficient of the cubic term must be non-zero in order to apply the aforementioned Landau potential to the first-order phase transition; otherwise,

Equation (4.1) is applicable only to a second-order phase transition or only at equilibrium where the two minima are equivalent. ζ represents the unstable hump for the crystal nucleation to overcome the barrier, and W is the coeﬃcient that represents the penalty for the nucleation process. ζ0 represents the crystal order parameter at

102 the solidiﬁcation potential for crystallization that may be treated as supercooling or crystal melting temperature dependent (Figure 4.1).

In principle, the stable solid may vary from unity at equilibrium to some

0 ﬁnite values of ζ0 depending on the supercooling or the melting temperature. At Tm, the free energy densities of the melt and the solid are equivalent, implying the coex-

0 istence of the crystal and melt. When T < Tm, the free energy density has a global minimum at ζ0 < 1; i.e., the linear crystallinity is less than unity, which suggests that the emerged crystals are defective or some amorphous materials are entrapped in the solidus phase reﬂecting the metastable nature of the polymorphous crystals.

Nonetheless, the metastable crystal phase is more stable than the unstable melt.

Hence, the melt must solidify by overcoming the nucleation barrier peak labeled by ζ on the y axis. As the supercooling increases, ζ0 moves to a lower value, which implies the imperfect crystal containing sizable amount of entrapped amorphous chains or defects. In the present study, only the equilibrium property is concerned and thus ζ0 is taken as unity. The advantage of the present theory of polymer solidiﬁcation is that these model parameters W and ζ are intimately related to the material properties of the individual components. This Landau-type free energy of solidiﬁcation has been successfully applied to describe the spatio temporal emergence of polymer single crystals, dendritic growth patterns, and dense lamellar branching in spherulites [34, 57].

Such non-equilibrium growth dynamics will not be discussed here as they are beyond the scope of the present work.

103 Figure 4.1: The variation of free energy of crystallization as a function of crystal order parameter, ψ, of a pure homopolymer, showing a symmetric double well at equilibrium between ψ = 0 and ψ = 1 representing the melt and the solid phase, respectively. The shape of free energy transforms to asymmetric having the crystal order parameter at the solidiﬁcation potential less than unity, reﬂecting the imperfect crystal (i.e., crystallinity of less than one) that may be attributed to the metastable nature of polymer crystallization.

104 4.3 Extension of the Phase Field Model of Crystallization to Polymer Blends

The total free energy density of mixing of a binary crystalline polymer blend may be expressed as the weighted sum of the free energy density pertaining to crystal solidiﬁcation of the crystalline constituent (f(ψ1) and f(ψ2) with its corresponding volume fraction (φ1 and φ2) and the free energy of liquid-liquid mixing as described by the Flory-Huggins theory of isotropic mixing [3] with the addition of the anisotropic interaction terms that account for crystalline-amorphous interaction and crystalline- crystalline interaction in that which follows:

φ (1 − φ) f(ψ , ψ , φ) =φf(ψ ) + (1 − φ)f(ψ ) + ln(φ) + ln(1 − φ) 1 2 1 2 r r 1 2 (4.2) 2 2 + {χaa + (χcaψ1 − 2χccψ1ψ2 + χacψ2}φ(1 − φ).

The ﬁrst and second term represent the Landau-type free energy of crystal solidiﬁ- cation of each component in which the individual free energy of the constituents is weighted by the respective volume fractions to ascertain that these potentials van- ish at the extreme limits of zero crystallinity or if a component is non-crystallizable.

The third and fourth term represents the entropy part of the free energy of mixing of the amorphous constituents. The ﬁfth term χaa corresponds to the amorphous- amorphous interaction parameter of Flory-Huggins that characterizes the stability of the liquid phase.

The anisotropic interactions, such as crystalline solid 1 - amorphous liquid 2 or amorphous liquid 1 - crystalline solid 2 interactions, may be denoted as χca and

105 χac, respectively. These anisotropic interactions of separate crystals and co-crystals are complimentary to χaa, representing isotropic interaction of amorphous materials.

Note that the order of the subscript in χ denotes the crystal or amorphous phase of

constituent 1 and of constituent 2, respectively, viz. χc1a2 ≡ χca , χa1c2 ≡ χac and

χc1c2 ≡ χcc . Moreover, these interaction parameters are proportional to the enthalpy of crystallization, i.e,

c c χca ∼ ∆H1/RT and χac ∼ ∆H2/RT (4.3)

c c where ∆H1 and ∆H2 are enthalpy of crystallization of components 1 and 2, respectively. Furthermore, the crystal 1 - crystal 2 interaction may be expressed as a geometric mean of the crystalline amorphous interactions (χca and χac) to ac- √ √ count for the non-ideal crystalline mixing, i.e., χc1c2 = cω12 χc1a2 · χa1c2 or √ √ χcc = cω χca · χac for simplicity, in which cω represents the anisotropic interaction parameter which signifies any departure from ideality (if it is different from unity). cω is associated with the anisotropic interactions only, which are intimately related to the heat of fusions of these individual crystals as well as of the co-crystals and thus it is not directly connected to χaa. The cω parameter and the geometric mixing rule is analogous to the ‘c’ anisotropic interaction parameter defined for the cases of smectic and nematic liquid crystal mixtures [41, 79] to measure the heat of mixing of the ordered phases and determine whether it is favorable or not.

106 Physically, the terms in the small bracket of Equation (4.2) can be explained as follows. By deﬁnition, the crystal order parameter ψ1 is the linear crystallinity of the component 1 and thus the product with its volume fraction ( ψ1φ ) corresponds to the bulk crystallinity in the blend. On the other hand, the product of (1 − φ) and ψ1 implies the amount of amorphous materials interacting with the crystalline phase, and hence the term χcaφψ1(1 − φ)ψ1 signiﬁes the crystalline-amorphous interaction. The same argument may be made to the second crystalline component, i.e.,

χacψ2(1 − φ)ψ2φ. The cross-interaction term, χccψ1φψ2(1 − φ), may be interpreted as the crystalline-crystalline interaction that occurs when crystals of φψ1 attempt to mix with (1 − φ)ψ2. Thus, χcc is a measure of the heat of mixing of the crystalline phases and determines whether it is favorable or not.

4.4 Construction of Phase Diagrams via Free Energy Minimization

In order to construct the phase transition points and determine the coexistence curves, we minimize the free energy ﬁrst with respect to all non-conserved order parameters ψ1 and ψ2 at every concentration φ, by taking the derivatives with respect to ψ1 and ψ2 using the Gauss-Newton method in conjunction with the Hes- sian matrix. The Gauss-Newton method of minimization is a recursive algorithm to minimize a multi variable function

n+1 n −1 ψi = ψi − [H (f)] ∇f (4.4)

107 where the gradient vector and the Hessian matrix is deﬁned as

   2 2  ∂f (ψ1, ψ2, φ) ∂ f (ψ1, ψ2, φ) ∂ f (ψ1, ψ2, φ)  ∂ψ2 ∂ψ ∂ψ  ∇f =  ∂ψ1  and H (f) =  1 1 2  (4.5)    2 2   ∂f (ψ1, ψ2, φ)   ∂ f (ψ1, ψ2, φ) ∂ f (ψ1, ψ2, φ)  2 ∂ψ2 ∂ψ2∂ψ1 ∂ψ2 where

∂f (ψ1, ψ2, φ) 0 = φf (ψ1) + φ (1 − φ) (2χcaψ1 − 2χccψ2) ∂ψ1

∂f (ψ1, ψ2, φ) 0 = (1 − φ) f (ψ2) + φ (1 − φ) (2χacψ2 − 2χccψ1) ∂ψ2 ∂2f (ψ , ψ , φ) 1 2 = φf 00 (ψ ) + φ (1 − φ) 2χ ∂ψ2 1 ca 1 . (4.6) 2 ∂ f (ψ1, ψ2, φ) 00 2 = (1 − φ) f (ψ2) + φ (1 − φ) 2χac ∂ψ2 2 ∂ f (ψ1, ψ2, φ) = −2χccφ (1 − φ) ∂ψ1∂ψ2 2 ∂ f (ψ1, ψ2, φ) = −2χccφ (1 − φ) ∂ψ2∂ψ1

In order to determine the crystal-liquid (melt) phase transition of the constituents, the minimization of the free energy was ﬁrst undertaken with respect to the individual crystal order parameters with a tolerance of 10−7 for various temperatures.

Subsequently, a common tangent algorithm was employed to construct the coexistence curves by setting the chemical potentials to be equal and subsequently seek the solutions to Equation (4.2) in what follows:

α β ∂f (ψ1, ψ2, φ) ∂f (ψ1, ψ2, φ) f (ψ1, ψ2, φ ) − f ψ1, ψ2, φ = = α β . (4.7) ∂φ φα ∂φ φβ φ − φ

108 Figure 4.2: The variation of the crystal order parameters of the constituents with volume fraction at various temperatures. The parameters utilized where c c r1 = 10 , r2 = 10 , cω = 0.65 , ∆H1 = 12kJ/mole, ∆H2 = 16kJ/mole, 0 0 Tm,1 = 500K, and Tm,2 = 480K. The results of the Gauss-Newton minimization show that below the melting temperature the crystal order parameter of both the components are coupled with each other. For temperatures 470 and 450 K, the order parameters slowly decrease as we proceed towards intermediate concentration and sharply falls to zero. This is characteristic of ﬁrst order transitions and indicates the possibility of crystalline-melt coexistence region. At 440 K, both the order parameters are fully developed suggesting that there is no crystalline-melt coexistence region though the possibility of a crystalline-crystalline coexistence still remains.

109 Figure 4.3: Free energy curves for a binary crystalline eutectic system (left) due to unfavorable solid-solid mixing as a function of descending temperature showing the formation of various coexistence regions such as crystal-melt and crystal-crystal. The parameters are the same as used in Figure 4.2. The corresponding free energy curves show that at 470K exist two crystalline-melt coexistence regions. At 450K, even though we see a crystalline-melt transition in Figure 4.2, the crystalline-melt coexistence might be less favorable than the crystalline- crystalline coexistence. At 440K, it is very clear that the melt state is no longer favorable and a crystalline-crystalline coexistence region exists.

110 4.5 Results and Discussion

When the crystalline mixture is completely immiscible in the solid crystalline state and forms separate crystals, the cross-interaction parameter can be taken as zero, i.e., cω = 0, and thus the crystal order parameters develop independently of each other which is a typical characteristic of a completely non-ideal solid solution.

On the other hand, if the system is a completely ideal solid solution, the crystalline mixture is completely miscible at all concentrations, i.e., cω = 1, and then the crystal order parameters must be coupled with each other so that the systems can undergo co- crystallization. In the case of intermediate situation when the binary crystalline mixture is partially miscible, e.g., cω = 0.65, a eutectic phase diagram can emerge where the two crystals are in equilibrium with the amorphous solvent. In the present case, a hypothetical mixture under consideration is composed of two crystalline constituents

c with the heat of crystallization of a hypothetical polymer pair ∆H1 = 12 kJ/mole and

c 0 ∆H2 = 16 kJ/mole, having the equilibrium melting temperatures of Tm,1 = 500K

0 and Tm,2 = 480K, respectively. The polymer mixture is symmetric with equal chain segments, say r1 = 10, r2 = 10 and the non-ideality (or partial miscibility) of the crystal-crystal mixture may be described via setting cω = 0.65. We first minimize the total free energies with respect to the crystal order parameters to determine the solid-liquid phase transition lines. As depicted in Figure 4.2, the crystal order parameters representing the crystal-melt transition drop off discretely at a certain volume fraction as typical for the first order phase transition. This phase transition line

111 advances toward the middle concentrations when the temperature is progressively lower and the system eventually approaches the eutectic point to be shown later in

Figure 4.4(c). Next we determine the coexistence curves by balancing the chemical potentials of the constituents in each phase. Figure 4.3 illustrates the variation of the free energies of the eutectic crystallization in which the free energy curves of the crystalline solids are intersecting with the single free energy well of the isotropic liquid

(470 to 450 K ). The points of intersections deﬁne the crystal-melting (amorphous) transitions of the constituents. These transition points shift progressively toward the middle concentrations in the descending order of temperature. Concurrently, one can discern the appearance of the dual free energy minima, representing the co-existence of two crystal solid phases (Figure 4.3). At 440 K, the two crystalline-amorphous phase transitions emerge to a eutectic point exhibiting the dual minimum free energy wells to suggest that the two crystalline solids are in equilibrium with the isotropic liquid (Cr-L-Cr). These free energy curves were utilized in the establishment of the eutectic-type phase diagrams. If the system were undergoing liquid-liquid phase separation, such a non-ideal liquid solution may be represented by a double-well. Figure

4.4(a) illustrates the solution with cω = 1 showing an ideal solid solution phase diagram with a crystal-liquid coexistence region bound by the liquidus and solidus lines. The value of cω < 1 implies that the crystallization is favored to occur as separate individual crystals and cω > 1 indicates the formation of azeotropic co-crystals, which will be discussed later. Figures 4.4(b) and 4.4(c) illustrate the development of

112 various topologies of phase diagrams with decreasing cω in which the crystal-crystal coexistence curve is pushed upward while the crystal-melt coexistence region is thrust downward (Figure 4.4(b)). At the same value of cω = 0.65 , the two curves intersect and form a tri-critical point which is also known as the eutectic point in the phase diagram (Figure 4.4(c)), displaying the crystal-melt and crystal-crystal coexistence regions. It should be emphasized that the present calculation was undertaken under the assumption that the isotropic melt phase is stable, i.e., χaa χcrit . One can contemplate the possible liquid-liquid phase separation in the liquid phase, through the variation of χaa . The increase in χaa broadens the solid crystal - solid crystal coexistence region. Concurrently, the eutectic point falls below the L+L, Cr+L, coexistence regions (Figure 4d), especially when χaa χcrit , i.e., the liquid - liquid coexistence gap protrudes above the crystal-liquid curves. As exempliﬁed in Fig- ure 4.4(d), the binodal curve for liquid-liquid phase separation is now located at a much higher temperature than the melting temperatures of either component. In the descending order of temperature, one can clearly discern a monotectic line (dotted line) between the L+Cr and L+L regions and also the eutectic line (dot-dashed line) dividing the L+Cr and Cr+Cr regions. A monotectic line is deﬁned as the crystal- liquid-liquid (Cr-L-L) coexistence line in the phase diagram where the system phase separates into a crystalline phase and two liquid phases. Such crystalline systems are of interest not only from the thermodynamic point of view, but also from the

113 growth dynamics of the microstructures that involve interplay between solidiﬁcation dynamics of crystal-crystal, crystal-liquid, and liquid-liquid demixing.

Another type of phase diagram that can be elucidated in the framework of the present model is the peritectic phase diagram. A peritectic line is deﬁned as the crystal-crystal-liquid (Cr-Cr-L) coexistence line as opposed to the crystal- liquid-crystal (Cr-L-Cr) line of a eutectic. The peritectic phase diagram is observed when the crystal-crystal coexistence curve intersects with the crystal-liquid curve of an ideal solid solution (Figure 4.5(a)). When the value of χaa increases beyond

χcrit while maintaining the ideality of the solid solution, i.e., cω = 1 , the crystal- crystal coexistence curve intersects with the crystal-liquid, forming a peritectic (Cr-

Cr-L) coexistence line. A further increase in the χaa value results in the appearance of the liquid-liquid coexistence envelope at a higher temperature than those of the crystal meting transition of the pure constituents. Concurrently, the monotectic line develops above the existing peritectic line. For the sake of completeness, an azeotrope phase diagram may be established for the case of ideal liquid solution but non-ideal solid solution as well as for the case of non-ideal liquid and non-ideal solid solution. The azeotrope is deﬁned as a blend in which the crystal solid solution (co- crystal) is favored in the mixed state than in their individual pure states (separate crystals). Such phases have been observed in the case of binary nematic mixtures and also for the binary smectic mixtures. Figure 4.6(a) shows the development of the azeotrope as a consequence of increasing the miscibility between the two-crystal

114 solid solution (cω > 1), which leads to the formation of an invariant point higher than the melting transition of either component. The crystal-crystal coexistence curve is now suppressed appreciably to a lower temperature and thus is not shown in the

Figure 4.6(a). As depicted in Figure 4.6b, the azeotrope can also be inﬂuenced by the amorphous-amorphous interaction parameter χaa . When χaa > χcrit , the liquid- liquid coexistence curve protrudes above the azeotropic point. This has led to the transformation of this azeotropic point to a new invariant point at which the crystal phase is in equilibrium with two liquid phases, i.e., the L-Cr-L line.

4.6 Comparison with Experiment

To justify some of the predictions afforded by the present theory, it is instruc- tive to compare the solutions with the experimental phase diagrams of the binary crystalline polymer blends, if available. We are fortunate to notice the intriguing phase diagrams reported by Smith and Manley [47] for two polyethylene fractions of different molecular weight, denoted 1,000 (PE1000) and 2,000 (PE2000), respectively. Thermal characterization and x-ray diffraction revealed the existence of a common crystalline lattice in which both constituents crystallize simultaneously. Figure 4.7(a) shows the single melting transitions of the two PE fractions as obtained by the DSC experiment, showing the systematic variation of the melting temperatures. It is not surprising to discern an ideal liquid solution since PE fractions contain the same PE molecules

∼ and χaa = 0 . In our calculation, we utilized the material parameters and experi-

115 Figure 4.4: Development of a eutectic in an ideal mixture. The solid-solid miscibility parameter becomes unfavorable to mixing and leads to the development of a eutectic: a) ideal crystal solution, cω = 1 ; b) non-ideal solid solution, cω = 0.85 ; and c) eutectic phase diagram at cω = 0.65 . d) When the non-ideality of liquid solution increases beyond the crystal-melt phase transition temperatures, i.e., χaa χcrit , a L+L coexistence curve appears above the Cr+L coexistence curves.

116 Figure 4.5: a) the phase diagram of an ideal liquid solution, but non-ideal solid solution with χaa ≥ χcrit leads to the formation of a peritectic (Cr-Cr-L) and b) the non-ideal liquid solution and non-ideal solid solution with χaa χcrit result in the formation of a monotectic (L-L-Cr) above the peritectic line.

117 Figure 4.6: a) Development of the azeotrope when solid-solid mixing becomes favorable such that the crystal phase is induced in the mixture at a higher temperature than the pure transition temperature of both the components. b) When the liquid-liquid mixing is non-ideal, i.e., χaa χcrit in the vicinity of the transition temperatures results in the formation of a L+L coexistence curve above the Cr+L coexistence curves.

118 c mental conditions reported by Smith and Manley [47], viz., ∆HPE2000 = 3.68kJ/mol,

c 0 0 ∆HPE1000 = 3.80kJ/mol, Tm,P E2000 = 378.2K, Tm,P E1000 = 393.7K, r1 = 70, and r2 = 128. The observed single melting transition indicates that the crystal solid solution must be ideal, and thus the cross-interaction term is taken as unity, i.e., ω12 = 1.

The two lines exhibiting convex and concave curvatures form the liquidus and solidus loop (Figure 4.7(a)). The experimental melting transition points are closer to the liquidus line. Although the solidus curve is identifiable in the calculated phase diagram, it is difficult to be detected experimentally due to the broad nature of the experimental DSC peaks affected by the close proximity to these solidus and liquidus lines.

Another interesting system investigated by Wittman and Manley [45] is the eutectic mixture of polycaprolactone (PCL) and trioxane. PCL employed in their study had a reported molecular weight of 14,000 with a melting temperature of

334K, whereas trioxane is a high melting solvent that has a melting temperature of 335K. In Figure 4.7(b) is shown the eutectic phase diagram of the PCL/trioxane mixture solved using the material and experimental parameters reported by Wittman

c c 0 and Manley, viz., ∆HPCL = 4.14kJ/mol, ∆Htrioxane = 3.4kJ/mol, Tm,P CL = 334K,

0 Tm,trioxane = 335K, r1 = 1, 400, and r2 = 1. It is assumed that the liquid solution is completely ideal, whereas the crystal solid solution is highly non-ideal, i.e.,

ω12 = 0.1. Wittman and Manley [45] solved these melting point depression curves in the context of the Flory diluent theory by treating one constituent at a time without

119 the interference of the other. Their calculated liquidus lines certainly captured the trend of the melting point depression, but the solidus lines were absent by virtue of the inherent complete immiscibility assumption in the original Flory theory. In the present case, the calculated liquidus lines exhibit remarkable match with the experimental melting transition points, except for the anomalous double peaks observed experimentally, which had been attributed to possible segregation by Wittman and

Manley [45]. Again, the calculated solidus lines are located very close to or coincide with the pure component axis, which are diﬃcult to obtain experimentally. It should be emphasized that it is not our intention to perfectly match our solutions with the reported phase diagrams because the equilibrium phase diagrams are rare to ﬁnd in polymer solutions and/or blends. Nevertheless, the present theory certainly captured the trends of the eutectics as well as the solidus lines which have been ignored in the crystalline polymer phase diagrams. We demonstrate the role of the crystalline- amorphous interactions and crystalline-crystalline interactions in the establishment of the eutectic, peritectic, and azeotropic phase diagrams of binary crystalline polymer blends, which are consistent with other materials phase diagrams such as metal alloys and liquid crystal mixtures.

4.7 Conclusions

Various phase diagrams including eutectic, peritectic, and azeotropic phase diagrams of a two-crystalline polymer blend have been established in the framework

120 of a phase ﬁeld theory by taking into consideration crystalline-amorphous interactions and crystalline-crystalline interactions. The solution of our combined free energies of liquid-liquid demixing and crystal solidiﬁcation potentials revealed the binary phase diagrams for two-crystalline polymer blends, consisting of liquid-liquid, liquid-solid, and solid-solid coexistence regions bound by the liquidus and solidus lines. The calculated eutectic phase diagrams were found to accord well with the experimental eutectic phase diagrams of polyethylene fractions as well as those of PCL/trioxane mixtures of Manley and co-workers [45,47].

121 Figure 4.7: a) Solid solution of polyethylene fractions in comparison with the experimental data of Smith and Manley [47], b) PCL/trioxane eutectic phase diagram [45] showing the presence of crystal-liquid and crystal-crystal regions bound by the liquidus and solidus lines.

122 CHAPTER V

DYNAMICS OF CRYSTALLIZATION IN CRYSTALLINE AND AMORPHOUS

BLENDS

5.1 Introduction

Much progress has been made over the past 50 years in understanding the dynamics of first order phase transitions [29]. Various macroscopic theories of crystallization for metal alloys [80] and polymers [13,14] have been proposed for predicting the experimentally measured quantities such as growth rate and percent crystallinity although determining the evolution of the microstructure of crystalline phases continues to be a challenge. Crystallization is thermodynamically termed a first order transition process that implies discontinuities in fundamental thermodynamic quantities such as internal energy and specific volume at the transition/melting temperature that requires explicit boundary tracking algorithms which are computationally expensive to implement. A breakthrough that has allowed researchers to obtain a complete spatial and temporal solution to such problems, which are also known as

Stefan type problems, has been through the use of order parameters [29, 31] to establish the thermodynamics of ﬁrst and second order transitions. Order parameters are additional thermodynamic variables that are continuous and vary smoothly al-

123 beit sharply around the transition point. The spatio temporal profiles of these order parameters are achieved by solving the thermodynamic model using the Time Depen- dent Ginzburg Landau (TDGL) equations of motion [81]. This whole methodology is also popularly referred to as phase field modeling. Its merits in the description of the solidification process in melts and solid solutions has been established in the past two decades by several prominent researchers [29,35,78,82]. Various interesting morphologies such as the structure of dendrites [83, 84], faceted single crystals [34], dense lamellar branching [57], eutectic [85, 86], doublon, seaweed [83], etc. seen in a variety of material systems such as organic/inorganic crystals, metal alloys, polymer melts have been determined using this approach.

Most binary systems involve constituents that are both capable of crystallizing but the majority of polymer blends are unique in contrast to metal alloys and eutectic due to a large number of polymer blends comprising of only one crystallisable constituent. The thermodynamics to explain the reduction in experimentally determined melting points in such blends has been proposed by Flory and referred to as the Flory diluent theory or the melting point depression relationship [6, 7]. A critical assumption is made in the Flory diluent theory that supposes the formation of neat/pure crystalline phases in a polymer blend. We have shown several lacunae that arise due to this assumption and proposed a more generalized model using the concept of order parameters that is able to encompass the Flory diluent theory. We have incorporated the concept of a crystalline-amorphous interaction parameter that

124 drives the solid-liquid phase separation and arrived at solutions that allow us to predict both the solidus and liquidus lines in polymer blends containing only a single crystalline species. In this chapter, we have solved the phase field model of crystallization extended to polymer blends [87] using the TDGL Model C equations. We have elucidated the mechanism of rejection as well as entrapment of the amorphous material that lead to the experimentally observed concentration profiles during crystallization in polymer blends. Also, we have studied the change in growth rate from linear to nonlinear as a consequence of the diffusive mass transport coupling with the crystallization growth rate and found some interesting results. Non-equilibrium effects predominate in polymer crystallization due to chain connectivity considerations, crystallization growth rate and diffusion coefficients that are orders of magnitude smaller in comparison with small molecule systems. A study of the spatial and temporal evolution of the concentration order parameter during solid-liquid phase separation yields us the different concentration profiles which elucidate the competitive mechanism between crystallization growth rate and diffusion that lead to the experimentally observed concentration profiles. In the following sections we will discuss the formulation of the Time Dependent Ginzburg Landau Model C (TDGL-C) equations and discuss the predicted profiles in comparison with some experimentally reported observations.

125 5.2 Review: Thermodynamic Model

The crystallization process is tracked using the crystal phase order parameter

ψ such that ψ = 0 denotes the melt state and ψ ≈ 1 denotes the solid state [35, 57,

78]. The crystal melt interface is of a small albeit ﬁnite thickness over which all thermodynamic quantities vary smoothly including ψ which varies from 0 < ψ < 1.

In the present context, the free energy of crystallization under quiescent conditions for a pure component is expressed using a free energy functional in ψ such that

Z ψ f = [flocal + fgrad]dΩ (5.1)

where flocal is the local free energy term and fgrad is the gradient free energy term. flocal is expressed as a Landau expansion in ψ as follows

Z Flocal(ψ) flocal(ψ) = = W ψ(ψ − ζ0)(ψ − ζ)dψ (5.2) NAkBT

where NA is the Avogadro constant, kB is the Boltzmann constant and T is the temperature of crystallization. As elucidtated in Figure 4.1, the local free energy has an asymmetric double well structure essential for describing ﬁrst order transitions.

W and ζ are supercooling dependent dimensionless constants that determine the free energy change upon crystallization and ease of nucleation. ζ0 represents the degree of perfection of the crystalline phase [57], equal to unity at the equilibrium melting

0 temperature of the pure crystalline constituent and less than unity for any T < Tm.

126 0 At the crystal melting temperature, Tm, the free energy densities for the crystalline

0 state and the melt are equivalent. When T < Tm, the free energy density of the crystal ordering has a global minimum at ψ = ζ0. The solid crystal phase is therefore a stable phase as it has the lowest free energy, and thus the melt is metastable. The free energy density associated with the formation of interfaces can be accounted for by the gradient free energy term expressed as follows

|κ ∇ψ|2 f (ψ) = ψ (5.3) grad 2

where κψ is the interface gradient energy coeﬃcient which can be taken as a scalar for simplicity [34].

In Chapter 3, we have extended the thermodynamic model of crystallization in conjunction with the Flory-Huggins free energy of mixing to describe the thermodynamics of crystallization in blends. A quick review is presented in this section for the sake of continuity. The concentration order parameter φ is deﬁned as the volume fraction of the crystalline constituent such that the total free energy of the blend can be written as

ψ ψ,φ φ ftotal = φf + f + f (5.4) where the ﬁrst term is a weighted addition of f ψ with φ that ensures that the model reduces to the pure component model in the limit of φ → 1. f ψ,φ is the demixing

127 energy density for solid-liquid phase separation and can be expressed as

ψ,φ 2 f = χcaψ φ(1 − φ) (5.5)

where χca is a dimensionless quantity that characterizes the extent of demixing during crystallization and establishes the solid and liquid compositions at equilibrium. f φ is the free energy of mixing of the amorphous mixture that can be expressed using the

Flory-Huggins theory of mixing [1,3] in conjunction with a gradient free energy such that 2 φ φlnφ (1 − φ)ln(1 − φ) |κφ∇φ| f = + + χaaφ(1 − φ) + (5.6) r1 r2 2

where r1 and r2 are the statistical segment lengths of the constituents and χaa is the amorphous-amorphous interaction energy density between the constituents. κφ is the interface gradient coeﬃcient for the non-local free energy density of the mixture [5] which is also treated as a scalar. The rigor of the total free energy can be ascertained

ψ by analyzing the limits of the free energy w.r.t. φ. As φ → 1, ftotal → f and the blend behaves as a single crystalline component by itself and as φ → 0, ftotal → 0.

Non-equilibrium effects predominate in polymer crystallization due to chain connectivity considerations, crystallization growth rate, and diffusion coefficients that are orders of magnitude smaller in comparison with small molecule systems. A study of the spatial and temporal evolution of the concentration order parameter during solid-liquid phase separation yields us the different concentration profiles that elu-

128 cidate the competitive mechanism between crystallization growth rate and diﬀusion that lead to the experimentally observed concentration proﬁles. In the proceeding sections we will discuss the formulation of the Time Dependent Ginzburg Landau

Model C (TDGL-C) equations and discuss the predicted proﬁles in comparison with some experimentally reported observations.

5.3 TDGL Model C Equations

For a conserved and non-conserved order parameter [35, 82], the model C equations can be written as ∂ψ δf = −Γ total (5.7a) ∂t ψ δψ

∂φ δf = ∇ · Γ ∇ total (5.7b) ∂t φ δφ

where Γψ is the mobility coefficient [78] for the crystallization process that is related to the viscosity or rotational diffusivity of the crystallisable constituent in the blend and Γφ is the mobility of the concentration field which is equivalent to the diffusion constant for the binary system and can be assumed to be independent of φ for the sake of simplicity.

Upon determining the functional derivative of the free energy with respect to each order parameter and subsequent non-dimensionalization using the dimensionless variables, τ = Γψt/κψ,x ˜ = x/κψ,y ˜ = y/κψandz ˜ = z/κψ, we obtain a set of equations,

129 ˜ Figure 5.1: 2D simulation of the model with Γφ = 1 andκ ˜φ = 1. The initial concentration of the mixture, φ = 0.9. The thermodynamic parameters are W = 15, ζ = 0.1, ζ0 = 1, χca = 3, r1 = 1, r2 = 1 and χaa = 0. a) ψ ﬁeld shows the formation of the crystalline phase where ψ = 1 is denoted by the white color and the melt is represented by ψ = 0 denoted by the black color (see colorbar) at τ = 40. b) 1 − φ ﬁeld shows the corresponding concentration map of the amorphous material. The darker regions inside the crystalline phase are poor in amorphous content. This amorphous material is excluded from the growing front of the crystalline phase seen as a white halo.

130 ∂ψ = −φ W ψ(ψ − ζ )(ψ − ζ) + 2χ (1 − φ)ψ − ∇˜ 2ψ (5.8a) ∂τ 0 ca

∂φ ˜ ˜ 2 1 + lnφ 1 + ln(1 − φ) = Γφ∇ − + χaa(1 − 2φ) ∂τ r1 r2 1 (ζ + ζ) ζ ζ + W ψ4 − 0 ψ3 + 0 + χ ψ2(1 − 2φ) (5.8b) 4 3 2 ca ˜ 2 2 ˜ 2 + |∇ψ| − κ˜φ∇ φ

˜ 2 ˜ ˆ ˆ ˆ where Γφ = Γφ/(κψΓψ),κ ˜φ = κφ/κψ and ∇ = i∂/∂x˜ + j∂/∂y˜+ k∂/∂z˜. Thus, the two ˜ dimensionless quantities, Γφ andκ ˜φ are the dimensionless kinetic parameters which will be varied in the numerical experiments to study the evolution of concentration profiles for a given set of dimensionless thermodynamic parameters. Numerical simulations have been carried out in 1D and 2D using a central difference scheme in time and space with a spatial discretization step of dx˜ = 1, dy˜ = 1 and a time discretization step of dτ = 0.001 which was found to be an optimal time step for stable solutions. The 1D simulation grid consisted of 200 grid points and the 2D simulation consisted of a 256 × 256 box, in which a single nucleation was triggered at the center using an artificial seed in the ψ field with a Gaussian profile. The thermodynamic parameters used in the simulation are listed in the Figure 5.1 caption.

5.4 Results and Discussion

A 2D simulation is able to capture the typical concentration proﬁles that are developed in and around a growing crystalline phase. Figure 5.1(a) captures a

131 Figure 5.2: 1D simulation of the model with the same thermodynamic parameters as used in Fig.5.1. The plots are concentration of the amorphous component ˜ vs distance for different values of Γφ = (a) 4 (b) 2 (c) 1 (d) 0.5 (e) 0.25 (f) 0.125 at time τ = 4. The crystalline phase was nucleated in the center of the grid. As the crystalline phase grows outward the amorphous component is rejected out as seen by the peaks in the concentration profiles. This region also corresponds to ˜ the crystalline phase and melt boundary (not shown in figure). Γφ 1 implies that redistribution of material is much faster than the crystalline phase growth ˜ rate. As Γφ decreases from (a) 4 to (b) 2, the rejection peak increases and the crystalline phase growth front has moved a smaller distance. Also the concentration of amorphous constituent inside the crystalline phase has increases. This ˜ trend continues in (c), (d) and (e). In (f) Γφ 1, the crystal growth front moves much more rapidly and the rejection peak is quite small as well as very little change between the concentration of the crystalline phase and the melt.

132 ˜ Figure 5.3: Growth rate at diﬀerent Γφ = 0, 0.125, 0.25, 1, 4, showing the transition from linear to nonlinear growth rates using the same thermodynamic parameters as used in Fig.5.1. Crystallinity is computed by integration of the ψ parameter over the entire volume and then normalizing it using the maximum R R ˜ possible value ( ψdV/ ζ0dV ). If Γφ = 0 no redistribution of amorphous component is possible. This growth rate is linear and similar to that observed for pure ˜ crystals growing from their melts. As Γφ = 0.125 increases the growth rate be- ˜ comes slower and nonlinear. It is slowest at about Γφ = 0.25 and then increases again as the redistribution becomes much faster than crystalline phase growth rate.

133 snapshot of the ψ field at τ = 40 while Figure 5.1(b) is the corresponding concentration map of the amorphous constituent which clearly shows that the crystalline phase is richer in the crystalline constituent than in the surrounding melt. The white circular ring surrounding the crystalline phase (black) in Figure 5.1(b) is a amor- phus rich region surrounding the growing crystal that has been well-documented in several experiments in binary blends of polymers containing a single crystallisable constituent [88,89]. The amorphous component is driven out of the crystalline phase and accumulates on the growing interface while simultaneously being diffused away into the melt blend. In order to properly understand this phenomenon it is conve- nient to conduct a 1D simulation and directly obtain the concentration profile of the amorphous constituent while a crystalline nuclei is growing. In these 1D simulations,

˜ we study the role of the non-dimensionless parameter Γφ which is a ratio of the diffusion coeﬃcient Γφ of the crystallisable constituent in the blend, the inverse of the viscosity of the system Γψ and characteristic length scale of the crystalline-amorphous interface κψ. In Figure 5.2, plots of the concentration of the amorphous component

(1 − φ) against dimensionless distance (˜x) are presented by systematically varying

˜ Γφ = 4, 2, 1, 0.5, 0.25, 0.125 starting from a 90:10 crystalline amorphous blend. Figure

5.2(a) shows the snapshot of the concentration ﬁeld of the amorphous constituent at

τ = 4. The crystalline nuclei was seeded in the center of the ﬁeld and the crystal grows outwards towards the boundaries. The crystal order ﬁeld is not shown but the crystalline growth front can be tracked by observing the rejection peaks in the

134 ˜ ﬁgure. A larger value of Γφ = 4 signiﬁes that the mass transport in the system occurs faster than crystallization. In such a case, the crystalline phase is very poor in the amorphous constituent and drives it outside its growth front that can be seen as

˜ rejection peaks. Some interesting features of varying the Γφ are observed relating to the size of the rejection peaks and the growth of the crystal growth front. As seen in

˜ Figures 5.2(a)-(e), as Γφ is decreased from 4, 2, 1, 0.5 and 0.25, the rejection peaks of the amorphous constituent get stronger and the speed of the crystal growth front is slower, implying that the coupled transport equations are strongly interfering with each other. The lack of mass transport driving force retards the accumulated amorphous constituent on the crystal growth front from diffusing into melt blend. This also causes the speed of the crystal growth front to significantly slow down. Another feature that can be seen in these figures is that the concentration of the amorphous constituent is progressively increasing. We have also noticed that the crystal order parameter of the crystalline phase in these system also drops a little bit, perhaps pointing to the imperfect crystallization taking place. A striking result that is ob-

˜ tained upon further reducing Γφ = 0.125 is shown in Figure 5.2(f). The crystalline phase instead of slowing down further has actually traversed a distance almost the same as in Figure 5.2(a). There is very little rejection observed at the crystal growth front as well as very little depletion of the amorphous component within the crystalline phase. The crystal order parameter (not shown in ﬁgure) is also markedly smaller than its equilibrium value. One possible explanation for this could be that

135 if the mass transport is significantly smaller than crystalline growth rate, the system chooses to crystallize into a metastable minimum that is not the actual equilibrium minimum. The concentration profile in Figure 5.2(f) is similar to the one obtained experimentally by Billingham and coworkers [90] through fluorescence microscopy measurements on spherulites growing in a blend of isotactic and atactic polypropylene(Figure 2.20). This system has also been studied by using small angle neutron scattering [91] and small angle X-ray scattering [92] that confirm the entrapment of the amorphous constituent on the scale of the size of the crystalline lamellae within the spherulites.

In the 1D simulations that have been carried out with a single nucleus, we can

R R determine the temporal evolution of crystallinity through the relation ψdV/ ζ0dV where R ψdV is the total crystallinity of the system over the entire volume of simu-

R lation and ζ0dV is the maximum possible value of the crystallinity of the system.

The crystal crystal growth rate can be found by taking the slope of crystallinity vs

˜ time. In Figure 5.3, we can clearly see that when Γφ = 0, no mass transport occurs in the system or crystallization proceeds much faster than mass transport. Crystalliza- tion is controlled solely through rearrangement kinetics characterized by a constant growth rate. The crystal growth front proceeds with a constant velocity, highest in

˜ magnitude compared to all other cases. As Γφ increases to 0.125, the crystal growth front slows down a little bit, though there is very little mass redistribution as has

˜ been seen in Figure 5.2 (f). As Γφ further increases to 0.25, a minimum in the growth

136 rate can be observed. This is the parameter region where the two equations are possibly maximally coupled and interfering with each other. Also, the sharp non- linearity in the growth rate is very apparent. The initial growth rate is fast and then it slows down to zero as the crystal growth approaches the system boundaries. As

˜ Γφ increases further the growth rate again rises up, as mass transport becomes faster than crystallization, the interference between the two transport equations is reduced.

The choice of the metastable state of the crystal order parameter and the velocity selection rule for the crystal growth fronts must be dependent on the thermodynamic and kinetic coeﬃcients. An exact analysis of these selection rules is a ﬁeld of study by itself and provides an incentive for future study of this problem.

5.5 Conclusion

The mechanism of rejection of amorphous material from crystalline phases or their entrapment within the crystalline phases has been investigated using the

˜ dimensionless parameter Γφ. The results of the 1D and 2D clearly indicate the effect of diffusion mass transport as compared to the crystal mobility coefficient. This plays a vital role in determining the amount of amorphous material that is trapped within the crystalline phase and the resultant concentration profile in the whole system.

We have established the diﬀerent concentration proﬁles that can be observed during single crystal growth and spherulitic growth. We have also determined the growth rates for a single nuclei and elucidated the typical linear growth rate observed in melts

137 as compared to the nonlinear growth rate observed in blends due to mass balance constraints and mass transport kinetics.

138 CHAPTER VI

PHASE DIAGRAMS AND MORPHOLOGY EVOLUTION IN PP/EPDM

BLENDS

6.1 Introduction

Polyolefins, one of the largest commodity polymeric materials in the market place, have been widely studied over six decades in respects of synthesis, structural, physical, and mechanical properties point of views. However, research on blends of polyolefins is rather scarce relative to their neat forms. One of the polyolefin blends that gained attention is thermoplastic polyolefins (TPO) due to the enhanced impact strength and toughness of polyolefins for automotive applications. A typical example is a blend of polypropylene (PP) and ethylene-propylene rubber (EPR). EPR has been incorporated into PP through reaction in batch reactors or physical blending.

The PP/EPR blends formed by mixing in the batch reactor are already phase separated and thus thermodynamics may not play a role, but the emerged structure and properties of physically blended ones may be affected by miscibility phase diagrams as well as by dynamics of phase separation. These rubber modified polyolefins greatly improve the toughness and impact strength of the composites due to the rubber inclusion whereas polyolefin constituents afford good tensile properties of the composites

139 and also melt processability. A certain functional group may be introduced to EPR to aﬀord chemical sites for crosslinking. Such crosslinking reaction further provides rubber-like network properties, but often it occurs at the expense of the reduction in melt processability. To circumvent such short comings, mineral or paraﬃnic oil have been added to serve as a means of controlling of the swelling properties of the blends.

Depending on the chemical structure of the functional groups, such reactive poly- oleﬁn blends are often known as thermoplastic elastomers (TPE) or thermoplastic vulcanizates (TPV). A classical example of TPE is the blend of polypropylene (PP) and ethylene propylene diene monomer (EPDM) [93–96]. Despite a slight reduction in the rigidity or stiﬀness, these PP/EPDM blends exhibit enhanced toughness and impact strength as well as good resistance to ozone and UV radiation without losing

flow properties. The addition of small amount of PP raises the modulus and tensile stress of iPP/EPDM as compared to neat EPDM, and thus it has been regarded as a stiffness modifier. On the other hand, adding small amounts of impact modi-

ﬁer such as EPDM improves the toughness and impact strength of PP/EPDM at a marginal loss in tensile strength of the PP. As can be expected, the mechanical and physical properties of PP/EPDM blends are intimately related to the internal phase separated domain structures of the constituent phases. The emerged structures vary from a sea-and-island type to a bi-continuous structure which may be governed by thermodynamic phase diagram if it exists as well as kinetics of thermally induced or reaction-induced phase separation. In addition, the crystalline phase of PP in

140 the blends has to be addressed in evaluating the blend performance which has been ignored in literature.

In practice, the melt blends of commercial-grade PP and EPDM were perceived to be completely immiscible which may be a consequence of melt blending conditions in given mixing equipment or driven by chemical reaction. To investigate the miscibility of a polymer blend, solution blending is preferred although it is usually not a favorite practice in most industrial settings. This immiscibility perception changed when a lower critical solution temperature (LCST) was ﬁrst reported for the iPP/EPDM solution blends; this LCST was located very close to the melting temperature of the neat iPP [54]. Moreover, the melting transition of iPP intervened in the

LCST phase diagram of iPP/EPDM blend, thereby complicating the phase behavior. In order to decouple LCST and melting behavior of iPP, Ramanujam et al. [55] used syndiotactic polypropylene (sPP) instead of iPP because sPP is known to have a lower crystal melting temperature relative to that of iPP. The authors found the existence of combined LCST and upper critical solution temperature (UCST) phase diagrams in the sPP/EPDM blend in which the melting transition of neat sPP is located in between the LCST and the UCST. The motivation of the present article is (i) to reconcile the diﬀering opinions of the complete immiscibility perception and the aforementioned complex liquid-solid phase diagrams of PP/EPDM thermoplastic elastomer systems and (ii) to elucidate the governing mechanisms of the spatiotemporal development of blend morphology involving the competition between the

141 phase separation dynamics and kinetics of crystallization. This chapter describes theoretical modeling and simulation on establishment of thermodynamic phase diagrams of PP isomers/EPDM and dynamics of thermal quench induced-phase separation and morphology development during crystallization of PP isomers. To substantiate the signiﬁcance of these phase diagrams of iPP/EPDM and sPP/EPDM, we have developed a model for a crystalline-amorphous polymer blend in order to predict all possible phase diagram topologies and to compare some of these predictions with the observed cloud point phase diagrams. It may be anticipated that the present theoretical approach is capable of reconciling the discrepancy between the above phase diagrams and the perceived immiscibility of the PP/EPDM blend.

6.2 FH Theory for UCST and LCST Systems

The Flory-Huggins (FH) theory [1] has been customarily employed to establishing phase diagram of binary amorphous-amorphous polymer mixtures. χaa is the

FH amorphous-amorphous interaction parameter determining the enthalpy contribution towards mixing [6, 24], which is proportional to the net interchange energy, but it is inversely proportional to absolute temperature. Note that the original FH free energy equation is only capable of predicting the UCST phase diagram of binary polymer blends. In order to account for free volume eﬀects involving non-ideality and non-combinatorial mixing, it is customary to express χaa in the context of an

142 empirical expression in what follows:

2 3 χaa = (A + B/T + C ln T ) Dφ + Eφ + F φ + ... (6.1)

where the ﬁrst bracket represents athermal and thermal dependencies, whereas the second bracket term accounts for all the concentration dependencies [67]. This modi-

fied model has been used extensively to determine a variety of phase diagram topologies such as UCST, LCST, a combined UCST-LCST, a closed loop, and/or an hour- glass phase diagram. Establishment of phase diagrams can be accomplished by applying the common tangent algorithm via the free energy minimization of the mixture [97] (see Appendices A and B). Although the FH theory may be adequate for elucidating the empirical phase diagrams of amorphous-amorphous, its extension to the crystalline-amorphous or crystalline-crystalline polymer blends requires a consid- erable modification by incorporating a solidification potential of the solid (crystal)

– liquid (melt) transition (i.e. crystallization) of the crystalline constituent(s) as described in previous chapters.

6.3 Prediction of Phase Diagram Topologies

In this section, we shall describe the various topologies of possible phase diagrams using the proposed thermodynamic model for a crystalline-amorphous blend.

Most polyoleﬁn blends exhibit either an LCST or a UCST or a combination of both. In order to model the combined LCST-UCST or hour glass phase diagram, the empirical

143 expression of χaa in Equation (6.1) may be treated only as a function of temperature, viz.,

χaa = A + B/T + C ln T (6.2)

Applying the conditions that χaa = χcrit at the critical temperature of the UCST and

LCST, one can treat these coeﬃcients to be a function of only a single adjustable parameter A, i.e.,

" 1 − 1 # B = (χ − A) ln(TUCST ) ln(TLCST ) (6.3) crit 1 − 1 TUCST ln(TUCST ) TLCST ln(TLCST ) and TUCST − TLCST C = (χcrit − A) (6.4) TUCST ln (TUCST ) − TLCST ln (TLCST ) where, 1 1 1 2 χcrit = √ + √ (6.5) 2 r1 r2

On the basis of Equations (6.3) – (6.5), we have solved for various phase diagram topologies of a hypothetical crystalline-amorphous polymer blend having a LCST temperature at 252oC and a UCST temperature at 202oC, but the melting transition temperatures and crystalline-amorphous interaction parameters vary. Figure 6.1 exhibits the inﬂuence of melting point on the phase diagram in columns from the top to the bottom as well as the eﬀect of the crystalline-amorphous interaction energy on the phase diagram in rows from left to right. As can be witnessed in Figure 6.1(a),

144 Figure 6.1: Hypothetical phase diagrams for crystalline-amorphous polymer blends exhibiting a combined LCST and UCST intersected by the crystal-melt transition gap bound by the liquidus and solidus lines, showing (a) eﬀect of melting temperature of the pure crystal component increasing from top to bottom (182, 223, and 262oC) and (b) the eﬀect of repulsive crystalline-amorphous interaction energy increasing from left to right. χca parameter is varied from 0.01, 0.1, and 1 at the melting temperature by setting the UCST temperature to 182oC and the LCST temperature to 262oC. The melting enthalpy of the constituent crystal was taken as 1500 cal/mole.

145 the coexistence of the LCST and UCST can be established in that the UCST is intersected by the crystal solid-liquid transition, displaying liquid-liquid, crystal-liquid coexistence regions, and the neat crystal gap shown by the solidus line at the high crystalline polymer concentrations. With increasing repulsive crystalline-amorphous interaction parameter we observe that the solidus line moves toward the pure crystal component axis and concurrently the neat crystal gap become narrower suggesting that more solvent is rejected out from the solidus phase (Figures 6.1(a)-6.1(c)). On the other hand, upon raising the melting transition temperature of the crystalline polymer constituent, the solidus region increases as the solidus line moves up (Figure 6.1(a) and 6.1(d)). With continued increase of the melting point, the crystal-melt transition is now intersecting with the LCST, thereby widening the crystal-liquid coexistence region like an hour glass phase diagram (Figure 6.1(g)). This kind of hour-glass phase diagram further transforms to almost completely immiscible crystalline solid – amorphous liquid (i.e., tree-trunk) gap with increasing repulsive crystalline-amorphous interaction (Figure 6.1(i)). These theoretical predictions indicate clearly that it is possible to discern intricate phase diagram topologies encompassing the LCST coexistence curve coupled with the crystal-melt transitions, the combined LCST/UCST phase diagram, and the solid-liquid coexistence region bound by the solidus and liquidus lines, all the way to the complete immiscibility of various PP/EPDM blends.

The present calculation strongly suggests that the intricate phase diagrams of the iPP/EPDM reported by Chen et al. [54] and that of sPP/EPDM by Ramanujam et

146 al. [55] are indeed possible and also the complete immiscibility perception for the melt blends of commercial PP/EPDM reported in literature is consistent with the present predictions.

6.4 Comparison with Experimental Phase Diagrams of PP/EPDM Blends

It is encouraging that diverse phase diagram topologies predicted by the present theory for the hypothetical crystalline-amorphous blend captures the observed trends of the phase diagrams of PP isomers/EPDM blends. It is essential to compare directly with the experimental phase diagrams reported by Chen and coworkers [54] for the iPP/EPDM blend (Figure 2.10) and the sPP/EPDM blend of Ramanujam et al. [55] (Figure 2.11) by utilizing the actual material parameters and the experimental conditions of iPP and sPP and their blends with EPDM. The material parameters utilized were the enthalpy of fusion of iPP, ∆HiP P = 2110cal/mol [98], and the statistical segment length, riP P = 1800 and rEPDM = 1000 , respectively. The equilibrium melting point of iPP is taken as 162.5oC and the LCST was calculated by setting A=0.01. The χca value was estimated to be 0.8 based on the heat of fusion of iPP, which also ﬁtted reasonably well with the experimental phase diagram at 162.5oC. Note that the density of the iPP crystal is approximated as unity so that the volume fraction roughly corresponds to the weight fraction of the PP. The conversion of the material parameters to the model parameters has been described in detail in the previous chapters (see appendices B and C).

147 Figure 6.2: Comparison between the calculated coexistence line and the cloud points (ﬁlled triangle) of iPP/EPDM and the melting points (ﬁlled circles). The solidus and liquidus lines are virtually overlapped (dots), but the existence of both lines is manifested by the kink in the LCST coexistence line. The phase diagram was calculated using the material parameters, ∆HiP P = 2110 cal/mol, o Tm = 162.5 C, riP P = 1800, rEPDM = 1000 and χca = 0.8 at melting temperature

148 Figure 6.3: Comparison between the calculated coexistence curves and experimentally observed LCST (open diamonds) and UCST spinodal gap (open diamonds) showing the liquidus (denoted by open circles) and solidus lines on the pure sPP axis. The material parameters utilized were the enthalpy of fusion o of sPP ∆HsP P = 1912 cal/mole, Tm = 123 C, rsP P = 1800, rEPDM = 1000 and χca = 0.8 at the melting temperature.

149 There are two possible scenarios to draw the phase diagram depending on the value of χca as demonstrated in Figure 6.1(g) and 6.1(i). With χca = 0.1, the solidus and liquidus lines that intersected with the LCST are coincided, which ﬁtted reasonably well with the way the melting transition points and the experimental cloud point phase diagram were drawn in the original paper. However, in view of the estimated χca = 0.8 value, the phase diagram seems more like that in Figure 6.1(i) with the solidus line being located right on the pure iPP ordinate. Figure 6.2 shows the comparison among the calculated coexistence curves and the experimental cloud points of the iPP/EPDM blends and the melting points of iPP in the blends. This observation suggests that some cloud point data falling below the liquidus line may be already in the crystalline solid-amorphous liquid coexistence region suggesting the non-equilibrium nature of the cloud point determination. This kind of theoretical prediction was not available at the time of the cloud point experiments by Chen and coworkers. To alleviate such a complex interplay between the liquid-liquid phase separation and the crystallization, one idea that was developed was to replace iPP with sPP by virtue of the lower melting temperature of sPP relative to that of iPP.

Upon replacing iPP with sPP, Ramanujam et al. were able to decouple the LCST of the sPP/EPDM with a minimum at around 150oC from the melting transitions of the sPP crystalline constituent located at 127oC (Figure 2.11). However, the sPP/EPDM blend exhibited the UCST peak at around 100oC which was buried under the melting point depression curve of sPP. The solution gives the combined

150 LCST/UCST phase diagram for this system by determining the model parameters using the material parameters for the sPP/EPDM blend. The material parameters utilized were the enthalpy of fusion of sPP ( ∆HsP P = 1912cal/mole), the equilibrium melting temperature for sPP was approximated as 127oC, and the statistical segment length, rsP P = 1800 and rEPDM = 1000 that roughly correspond to the molecular weights of the constituent polymers. The value was calculated through Equations

(6.3)–(6.5) by setting A = 0.01. Again the density of the sPP crystal is taken as unity so that the volume fraction in the theoretical description and the weight fraction of the experiment can be used interchangeably. The model parameter χca is estimated to be around 0.8 at 127oC from the ﬁt of the solution to the experimental cloud point curves.

As shown in Figure 6.3, the coexistence line afforded by the solution shows a good fit with the LCST cloud points. The calculated liquidus line also closely matches the crystal melting curve which is found to situate between the LCST coexistence curve and the UCST spinodal gap. The solidus line is situated right on the pure sPP crystal axis. However, the UCST coexistence line cannot be discerned since it has merged with the liquid and solidus lines, leaving the spinodal gap representing the unstable liquid envelope that was buried underneath the liquidus line. The tail end of the liquidus line curving downward asymptotically to the EPDM axis manifests the influence of the UCST, which in turn indicates that the UCST is merged with the liquidus line. At 67oC, it can be noticed that the buried spinodal spans all the way

151 to 10 wt% of sPP. This probably explains the occasional observation of the spinodal type phase separated domains found in the 10/90 sPP/EPDM blend that experience deep quenches at round 27 ∼ 67oC.

6.5 Dynamics of Crystal Growth in a Phase Separating System: iPP/EPDM Blends

In view of the complex phase diagram of iPP/EPDM blend, it can be anticipated that crystallization kinetics and morphology development in iPP/EPDM blends would be complicated by the aforementioned phase separation [53, 99–104].

However, the thermodynamic phase diagram depicted in Figure 6.2 certainly serves as a road map for the trajectory of thermal jump experiments. We use the same

Model C equations as derived in Chapter 5 to study the dynamics of crystallization and phase separation.

∂ψ = −φ W ψ(ψ − ζ )(ψ − ζ) + 2χ (1 − φ)ψ − ∇˜ 2ψ (6.6a) ∂τ 0 ca

∂φ ˜ ˜ 2 1 + lnφ 1 + ln(1 − φ) = Γφ∇ − + χaa(1 − 2φ) ∂τ r1 r2 1 (ζ + ζ) ζ ζ + W ψ4 − 0 ψ3 + 0 + χ ψ2(1 − 2φ) (6.6b) 4 3 2 ca ˜ 2 2 ˜ 2 + |∇ψ| − κ˜φ∇ φ

˜ 2 ˜ ˆ ˆ ˆ where Γφ = Γφ/(κψΓψ),κ ˜φ = κφ/κψ and ∇ = i∂/∂x˜ + j∂/∂y˜ + k∂/∂z˜ Thus, the two ˜ dimensionless quantities, Γφ andκ ˜φ are the dimensionless kinetic parameters. The

2-D simulation has been carried out for the 50/50 iPP/EPDM blend (i.e., a near

152 critical composition) following a temperature jump from a single phase temperature of 142oC to a temperature of 155oC which is below the crystal melting temperature, but it is above the LCST. The thermodynamic parameters used for the iPP/EPDM simulation are W = 10 , ζ = 0.1, ζ0 = 0.98, χca = 0.1,κ ˜ψ ∼ 1,κ ˜φ ∼ 1, r1 = 18,

˜ r2 = 10 and χaa = 0.25. We have chosen the kinetic parameter Γψ ∼ 10 to reﬂect the fast crystallization rate as seen in iPP melt crystallization studies. The numerical simulations were carried out on several grid sizes of 128 × 128, 256 × 256, and 512

× 512 with varying time steps of 0.001, 0.0005, and 0.0001 to ensure the stability of the simulation. The results of the 512 × 512 simulations are shown in Figure

6.4, exhibiting the emerged bi-continuous structure which is seemingly driven by liquid-liquid phase separation through spinodal decomposition in the composition

ﬁeld (upper row). The spinodal process is known to be spontaneous, thus any unstable

ﬂuctuation can grow very rapidly as this SD process does not require any energy to overcome. As can be seen in the bottom row, the nucleation (solid-liquid) of spherulite develops rather late relative to the phase separation, but it catches up very rapidly and eventually out-grows the SD domains. The spherulitic boundary can be discerned in the enlarged picture, which is out-growing over the interconnected SD domains.

Such knowledge was unavailable at the time of the experiments performed some ten years ago even though the observations were clearly hinting towards it. Although the experiment is not ideal for quantitatively comparison with the present simulation

153 regarding the competition between the phase separation and crystallization, it is still worthwhile to revisit that which was observed experimentally.

Figure 8 shows the development of spherulitic morphology in the 50/50 iPP/EPDM blend upon cooling from 230oC to ambient at a slow cooling rate of

0.5oC/min. As depicted in Figure 6.5(a), the polarized optical micrograph under the cross-polarizers clearly reveals the maltese-cross pattern indicative of the iPP spherulite structure. The corresponding light scattering study exhibit a four-lobe clover pattern in the horizontal-vertical (Hv) configuration, which further confirms the existence of the spherulite texture (Figure 6.5(b)). In the unpolarized configuration, the structure in the optical micrograph (Figure 6.5(c)) is reminiscent of the phase separated interconnected SD morphology. The corresponding light scattering under the vertical-vertical (Vv) configuration in Figure 6.5(d) reveals a large scattering halo suggestive of domination by the concentration fluctuations of the phase separated domains over the orientation fluctuations. That is to say, the orientation

ﬂuctuation can be attributed to the emerged crystalline spherulitic morphology the size of which is so large that it primarily contributes to the main beam. Nevertheless, the observed over-growth of spherulitic structure on the existing interconnected SD structures can be identiﬁed which accords very well with the above simulated structure. Hashimoto et al. [105] have also shown in a similar experiment the formation of these morphologies by quenching the system from a high temperature two phase

154 Figure 6.4: Temporal evolution of the crystalline microstructure in the 50/50 iPP/EPDM blend, following a T-quench from the isotropic melt to a supercooled temperature below both the UCST spinodal gap, showing the growth of spherulitic front in the concentration ﬁeld. However, the over-growth of this spherulitic boundary on the bi-continuous SD domain structures can be seen clearly only in the enlarged version.

155 Figure 6.5: Morphology development in a 50/50 iPP/EPDM during cooling from 230oC to room temperature at a slow cooling rate of 0.5oC/min was studied by Chen et al. [54]. a) Polarized optical micrograph under cross polarizers clearly showing the maltese cross pattern indicative of the spherulite structure. b) Four-lobe clover leaf pattern in SALS in the Hv configuration confirming the existence of the spherulite texture. c) Polarized optical micrographs under parallel polarizers showing the phase separated morphology and d) the ring pattern in the Vv configuration dominated by the concentration fluctuations of the phase separated domains.

156 state to a temperature that is lower than the crystal melting temperature yet above the LCST temperature for the system.

6.6 Dynamics of Crystal Growth in a Phase Separating System: sPP/EPDM Blends

In Figure 6.6 is shown the morphology development in a 50/50 sPP/EPDM blend isothermally quenched from 128oC (a single phase) to 100oC (two-phase under the UCST) for a prolonged period. The samples were analyzed under both unpolarized and cross-polarized conﬁgurations as time progressed. Even though phase separation occurs at this temperature, the cross-polar micrographs show the occurrence of crystallization, i.e., the PP rich domains are stretched along the bi-continuous regions dictated by the SD structure. A similar temperature quench has been carried out on the 70/30 sPP/EPDM blend which shows a diﬀerent trend. As evident in

Figure 6.7, the EPDM rich phase is dispersed in the form of globular droplets within the matrix of the PP rich phase. Thus the sPP crystals grow along channels by mean- dering around these EPDM rich droplet domains, forming EPDM islands in the sea of sPP crystalline continuum. It may be inferred that in both the 50/50 and 70/30 blends the crystal growth of sPP is conﬁned within the phase separated microstructure. This correlation between the crystalline and phase separated structures may be attributed to the strong crystalline-amorphous interaction that was predicted in the theoretical phase diagram and the slower crystallization kinetics of s-PP. A similar

157 Figure 6.6: Optical micrographs obtained for a 50/50 sPP/EPDM blend isothermally quenched at 100oC by Ramanujam et al. [55]. Left column depicts the evolution of phase separation under the unpolarized condition and right column indicates the growth of crystals under the cross-polarized condition.

158 Figure 6.7: Optical micrographs obtained for a 70/30 sPP/EPDM blend isothermally quenched at 100oC by Ramanujan et al. [55]. The feft column depicts the evolution of phase separation under the unpolarized condition, and the right column indicates the growth of crystals under the cross-polarized condition.

159 result has also been reported by Crist and Hill [106] during thermal quenching of polyethylene and hydrogenated polybutadiene near the critical concentration.

A 2D simulation of the kinetics of phase separation and morphology evolution in sPP/EPDM blends conﬁrms this conjecture. Initially the system was set at a temperature (127oC) which is slightly higher than the melt temperature but lower than the LCST temperature of the blend. First a thermal jump was carried out from

127oC to 157oC that lies inside the spinodal envelope of the LCST. No crystallization of the PP component was involved during such a jump. The morphology evolution is seemingly governed by the simple liquid-liquid phase separation. As shown in

Figure 6.8(a), the formation of the spinodal microstructures can be witnessed in the concentration order parameter ﬁeld. As phase separation continues the coarsening of the phases takes place as evidenced by the increase in periodic wavelength of the phase separated bi-continuous domains (Figure 6.8(b)).

Another experiment was carried out by quenching the 50/50 blend from the initial temperature of 127oC (i.e., the single phase) to a temperature of 77oC which lies underneath the liquidus line and the UCST spinodal. The same set of parameters for the iPP/EPDM blend was employed for the thermodynamic parameter χca = 0.8

˜ to account for the buried UCST immiscibility gap and the kinetic parameter Γψ ∼ 0.1 to reﬂect the slower crystallization of sPP relative to iPP. Nucleation was triggered by thermal noise in the crystal order parameter ﬁeld. During the initial stages, the formation of the spinodal phase separated structures can be witnessed in the

160 concentration ﬁeld, but the crystals have yet to emerge in the crystal order parameter

ﬁeld (Figure 6.9). With the progression of time, some nuclei formed in the PP rich regions because the probability of the nuclei to survive is much greater than in the PP poor regions. It can be envisaged that the growing crystalline regions are strongly dictated by the spinodal template, thereby loosing the radial growth habit such as spherulitic growth. This observation is indeed what was observed in the actual experiment of the sPP/EPDM blends (Figure 6.6).

In Figure 6.10, the simulated structures are shown for the 70/30 sPP/EPDM mixture under the same conditions as described in the preceding case. The asymmetry in the composition results in the change of mechanism of phase separation from the spinodal to the nucleation-growth mechanism. Now the minority EPDM-rich phase forms the droplets in the continuum of sPP-rich phase. As can be anticipated, the PP crystallization lags behind the liquid-liquid phase separation because phase separation must occur ﬁrst in order for the sPP phase to reach or exceed its critical concentration so that the crystal can nucleate. The crystallization of sPP in the blends proceeds by weaving around the discrete EPDM domains.

On the contrary, the sPP forms the droplets in the continuum of EPDM in the case of 30/70 sPP/EPDM. It is striking to discern that sPP crystals are strictly conﬁned in the sPP-rich droplets (Figure 6.11). The crystallization of sPP lags behind the liquid–liquid phase separation occurring through the nucleation and growth mechanism except that the former is governed by the solid-liquid phase transition as

161 opposed to the latter case of liquid-liquid phase separation. Again, the sPP concentration must reach the threshold value for the nucleation to occur which can only be achieved through phase separation.

6.7 Conclusions

We have demonstrated the interplay of solid-liquid phase separation and liquid-liquid phase separation in the blends of iPP/EPDM and sPP/EPDM, showing the influence of PP tacticity on phase diagrams. In the establishment of the experimental phase diagrams for the iPP/EPDM blend, we observed the intersection of the solid-liquid coexistence curves with the liquid-liquid coexistence curves which prompted the study of the sPP/EPDM blend to decouple the two competing processes. We have developed a thermodynamic model based on the crystalline- amorphous interaction in addition to the conventional amorphous-amorphous interaction of the FH theory. With this modification, one can predict various phase diagrams of crystalline-amorphous polymer blends exhibiting both LCST and UCST behavior coupled with the melting transition of one of the components. The crystal- liquid gap was bound by the solidus and liquidus lines which cannot be achieved by the original Flory diluent theory. Based on the present modified free energy, the interplay between phase separation and crystallization can be explicable based on the

TDGL-Model C equations of motion for the concentration and crystal order param-

162 eters. Various morphological features have been simulated that are consistent with the observed crystal morphologies of iPP/EPDM and sPP/EPDM blends.

163 Figure 6.8: Emerged bi-continuous structure following a temperature jump from the isotropic melt between the LCST and the melting transition into the LCST gap, which is presumably driven by liquid-liquid phase separation through spinodal decomposition in the 50/50 sPP/EPDM mixture: (a) 1000 s and (b) 3000 s

164 Figure 6.9: Competition between the liquid-liquid phase separation through spinodal decomposition and the crystalline structure formation in the 50/50 sPP/EPDM blend. The crystallization occurs with the preformed SD networks. The top and bottom rows represent the temporal evolutions of the concentration ﬁeld and the corresponding crystal order ﬁeld.

165 Figure 6.10: Competition between the liquid-liquid phase separation through nucleation and growth showing the droplet domains and the crystalline structure formation in the 70/30 sPP/EPDM blend. The crystalline sPP component being the major phase, the sPP crystallization occurs in the matrix by weaving around the EPDM domains. The top and bottom rows represent the temporal evolutions of the concentration ﬁeld and the corresponding crystal order ﬁeld.

166 Figure 6.11: Competition between the liquid-liquid phase separation through nucleation and growth and the crystalline structure formation in the 30/70 sPP/EPDM blend. The crystallization of sPP is confined to the sPP-rich droplets. The top and bottom rows represent the temporal evolutions of the concentration field and the corresponding crystal order field.

167 CHAPTER VII

CONCLUSIONS AND RECOMMENDATIONS

7.1 Conclusions

In this work, the phase field theory of solidification as applied to homopolymer crystallization has been extended for the determination of phase diagrams of a crystalline-amorphous polymer blend. Using the Flory diluent theory as the starting point, a modification was made to the complete immiscibility assumption of the polymeric solvent in the neat solid crystal by taking into consideration the crystalline- amorphous interaction. This approach allows the amorphous material to coexist within the crystalline phase. The model consists of a free energy of solidification based on the Landau expansion of the crystal order parameter (ψ) and the free energy of mixing based on Flory-Huggins theory of mixing using the concentration order parameter (or volume fraction φ). Due to the phenomenological nature of the model, we investigated the possible phase diagram topologies that could be generated upon varying the crystalline-amorphous interaction in addition to the amorphous- amorphous interaction of the pair. It was found that the complete immiscibility assumption in the original Flory diluent theory may result in over-predicting the χaa value. An attempt was made to calculate the phase diagram of the PVDF/PMMA

168 blend. Although the extremely complicated melting behavior of the PVDF/PMMA blend made it diﬃcult to judge the merits of the new approach based on the calculated phase diagram, there was a good agreement for a partially miscible system

(PCL/PS) exhibiting UCST behavior in conjunction with a melting transition. In such a case, the present theory permits the simultaneous determination of χca and

χaa from the melting transition and the UCST envelope, respectively. The calculated phase diagrams of crystalline-amorphous polymer blends were able to capture both solidus and liquidus lines forming monotectic phase diagrams consistent with those of other systems such as metal alloys, organic crystals and liquid crystals. It is also shown that the original Flory diluent theory is recovered at the limit of complete immiscibility assumption of solvent and solute crystals.

Binary polymer crystalline blends have traditionally been treated as crystalline- amorphous blends. Even though there has been evidence of the formation of eutectic phase diagrams in polymer blends, the concept of limited miscibility of crystalline mixtures has been ignored. We have extended the model for crystalline-amorphous blends to crystalline-crystalline blends by including crystalline-crystalline interactions. We have been able to demonstrate the formation of ideal solid solution phase diagrams and eutectic phase diagrams based on the crystalline-crystalline interaction terms. Peritectic and azeotropic phase diagrams have also been predicted using this model. The solution of our combined free energies of liquid-liquid demixing and crystal solidiﬁcation potentials revealed the binary phase diagrams for two-crystalline

169 polymer blends, consisting of various coexistence regions such as liquid-liquid, liquid- solid, and solid-solid coexistence regions bound by the liquidus and solidus lines. The calculated eutectic phase diagrams were found to accord well with the experimental phase diagrams of polyethylene fractions as well as that of PCL/trioxane mixtures of

Manley and co-workers [45,47].

We have used the proposed model for crystalline-amorphous polymer blends to study the spatial and temporal dynamics of crystallization using the time-dependent

Ginzburg Landau approach. The model consisted of two coupled partial differential equations pertaining to the spatio-temporal evolution of the crystal order parameter and the concentration order parameter. Upon non-dimensionalization, we studied the effect of a dimensionless transport quantity that reflects the ratio of diffusion mass transport and the speed of crystallization. Using 1D and 2D simulations, several interesting findings were made regarding the non-equilibrium nature of crystallization in blends. The rejection of amorphous material from crystalline phases or their entrapment within the crystalline phases was observed in the simulations as a result of the competition between crystallization and diffusion mass transport. It was observed that the interference of the two coupled processes can slow the overall crystallization rate. One of the striking results showed that when the mass transport is much slower than crystallization, the system goes to a non-equilibrium metastable state to minimize its energy within the prescribed time scale. We have also determined the growth rates for a single nucleus and elucidated the typical linear growth

170 rate observed in melts as compared to the nonlinear growth rate observed in blends due to mass balance constraints and mass transport kinetics.

Lastly, we studied the problem of structure evolution in thermoplastic elastomer blends of PP/EPDM. We have demonstrated the interplay of solid-liquid phase separation and liquid-liquid phase separation in the blends of iPP/EPDM and sPP/EPDM showing the inﬂuence of PP tacticity on phase diagrams. We used an empirical form of the amorphous-amorphous interaction parameter (χaa) in conjunction with the present model for crystalline-amorphous blends to describe phase diagrams exhibiting both a LCST and UCST. Based on the present model, the interplay between phase separation and crystallization was elucidated based on the TDGL-Model

C equations of motion for the concentration and crystal order parameters. Several of the morphological features such as crystal over run on phase separated templates observed in iPP/EPDM and crystal trapping within phase separated domains in sPP/EPDM blends were simulated using the Model C equations.

7.2 Recommendations

Several interesting ideas have emerged as a consequence of this work. One of them is the concept of spinodals and their physical meaning in the context of crystallization. It is possible to construct spinodals using the models developed in this work. Even in phase separation of amorphous mixtures, spinodals remain a hot topic of debate and whether they really depict a clear demarcation in the two types of

171 phase separation mechanisms (spinodal decomposition and nucleation and growth) is suspect. Detailed kinetics need to be run in order to understand the physical meaning of these boundaries that can be easily constructed mathematically to deco- rate phase diagrams. The problem of crystal structure transformations has not been discussed in this dissertation. Recently Matsyuma [12] has developed a model for

BCC to FCC transitions in crystalline mixtures. He has shown that it is possible to use multiple order parameters for describing the free energy of each crystal structure and to couple them to describe crystal-crystal transitions. It is possible to incorporate this phenomenon into the present approach as well. Other ongoing research has been in studying the effects of temperature and concentration simultaneously on crystallization in blends. This involves solving three coupled partial differential equations pertaining to crystallization, concentration and temperature order parameters for crystalline-amorphous blends. We used Equation (2.73) in conjunction with the equations of motion developed in Chapter 5. Initial work suggests that the internal structure of dendrites and spherulites could be influenced by concentration effects.

As seen in Figure 7.1, we used a six-fold anisotropy simulation to demonstrate the diﬀerent growth habits that are observed in the dendritic growth habits of a pure component and a crystalline-amorphous blend. In the results of the pure component dendrite simulations it is observed that the primary and secondary branches are very well-deﬁned in their growth axes [57] and the crystal structures are sharply faceted as shown by the simulation result (on the top) in Figure 7.1 (a). On the other hand,

172 in the presence of concentration eﬀects in crystalline-amorphous blend, even for a strong anisotropy, we observe that the branches are more winding and less correlated with the growth axes. The faceted feature is still preserved as shown in the (bottom) simulation result in Figure 7.1 (b). Dynamics of crystallization in binary crystalline blends whose thermodynamics have been developed in Chapter 4 could also be undertaken to elucidate the mechanism of eutectic formation that could possibly be used to construct refractive index gradient structures through self-assembly on the micron scale. Several interesting ﬁndings remain to be found which can be pursued in the future.

173 Figure 7.1: Comparison of two dendritic growth simulations. Top is the dendritic growth in a pure component and bottom is the dendritic growth in a crystalline-amorphous blend. The pictures are the surface maps of the crystal order parameter ψ.

174 BIBLIOGRAPHY

[1] P. J. Flory. Principles of Polymer Chemistry. Cornell University Press, N.Y., 1953.

[2] P. J. Flory. The Conﬁguration of Real Polymer Chains. J. Chem. Phys., 17:303, 2004.

[3] P. J. Flory. Thermodynamics of High-Polymer Solutions. J. Chem. Phys., 10:51–61, 1942.

[4] M. L. Huggins. Theory of Solution of High Polymers. J. Am. Chem. Soc., 64:1712–1719, 1942.

[5] J. W. Cahn and J. E. Hilliard. Free Energy of a Nonuniform System. I. Inter- facial Free Energy. J. Chem. Phys., 28:258–267, 1958.

[6] P. J. Flory. Thermodynamics of Crystallization in High Polymers. IV. A Theory of Crystalline States and Fusion in Polymers, Copolymers, and their Mixtures with Diluents. J. Chem. Phys., 17:223–240, 1949.

[7] P. J. Flory. Thermodynamics of Crystallization in High Polymers. II. Simpliﬁed Derivation of Melting-point Relationships. J. Chem. Phys., 15:684, 1947.

[8] L. A. Kleintjens, M. H. Onclin, and R. Koningsveld. Thermodynamics of Poly- mer Blends. EFCE Publication Series, 11:521–533, 1980.

[9] W. Maier and A. Saupe. A Simple Molecular Statistical Theory of the Nematic Crystalline-Liquid Phase. I. Zeitschrift fuer Naturforschung, 14a:882–889, 1959.

[10] W. Maier and A. Saupe. A Simple Molecular-Statistical Theory of the Ne- matic Crystalline-Liquid Phase. II. Zeitschrift fuer Naturforschung, 15a:287– 292, 1960.

175 [11] W. L. McMillan. Simple Molecular Model for the Smectic A Phase of Liquid Crystals. Phys. Rev. A: At., Mol., Opt. Phys., [3]4(3):1238–1246, 1971.

[12] A. Matsuyama. Spinodal in a Liquid–Face-Centered Cubic Phase Separation. Journal of the Physical Society of Japan, 75(8):084602–084602, 2006.

[13] J. I. Lauritzen and J. D. Hoﬀman. Extension of Theory of Growth of Chain- folded Polymer Crystals to Large Undercoolings. J. Appl. Phys., 44:4340–4352, 1973.

[14] J. I. Lauritzen and E. Passaglia. Kinetics of Crystallization in Multicompo- nent Systems: II Chain-folded Polymer Crystals. J. Res. Nat. Bur. Std. (US), 71A:261–275, 1967.

[15] J.D Hoﬀman, L. J. Frolen, and J. I. J. I. Lauritzen. On Growth of Spherulites and Axialites from the Melt in Polyethylene Fractions: Regime I and Regime II Crystallization. J. Res. Nat. Bur. Std. (US), 79A:671–691, 1967.

[16] J. D. Hoﬀman. Theoretical Aspects of Polymer Crystallization with Chain Folds: Bulk Polymers. J. Soc. Plas. Eng., 4:315–362, 1964.

[17] J. D. Hoﬀman and J. L. Lauritzen. Formation of Polymer Crystals with Folded Chains from Dilute Solutions. J. Chem. Phys., 31:1680–1681, 1959.

[18] J. D. Hoﬀman and J. L. Lauritzen. Crystallization of Bulk Polymers with Folding: Theory of Growth of Polymer Spherulites. J. Res. Nat. Bur. Stand., Sect. A, 65:297–336, 1961.

[19] H. D. Keith and F. J. Jr. Padden. Spherulitic Crystallization from the Melt. I.Fractionation and Impuritiy Segregation and their Inﬂuence on Crystalline Morphology. J. Appl. Phys., 35:1270–1285, 1964.

[20] H. D. Keith and F. J. Jr. Padden. Spherulitic Crystallization from the Melt. II. Inﬂuence of Fractionation and Impurity Segregation on the Kinetics of Crys- tallization. J. Appl. Phys., 35:1286–1296, 1964.

[21] M. L. Mansﬁeld. Computer Simulation Solution to the Frank Equation with Moving Boundary Conditions. Polym. Comm., 31:283–285, 1990.

176 [22] R. Mehta. Eﬀect of Self-generated Mechanical Field on Spatio-temporal Growth of Polymer Single Crystals, Hedrites, and Spherulites During Isothermal Crys- tallization. PhD thesis, University of Akron, Dept. of Polymer Engineering, 2003.

[23] H. Xu. Eﬀect of Self-generated Thermal Field on Spatio-temporal Growth of Hierarchical Structures During Polymer Crystallization. PhD thesis, University of Akron, Dept. of Polymer Engineering, 2004.

[24] J. M. Prausnitz, R. N. Lichtenthaler, and E. G. Azevedo. Molecular Thermo- dynamics of Fluid-Phase Equilibria. Prentice-Hall, N.Y., 1999.

[25] S. P. Tan and M. Radosz. Gibbs Topological Analysis for Estimating Phase Equilibrium Tie Lines. Fluid Phase Equil., 216:159–165, 2004.

[26] T. Nishi and T. T. Wang. Melting Point Depression and Kinetic Eﬀects of Cool- ing on Crystallization in Poly(vinylidene ﬂuoride)-Poly(methyl methacrylate) Mixtures. Macromolecules, 8(6):909–915, 1975.

[27] W. R. Burghardt. Phase Diagrams for Binary Polymer Systems Exhibiting both Crystallization and Limited Liquid-Liquid Miscibility. Macromolecules, 22(5):2482–2486, 1989.

[28] G. Caginalp and P. Fife. Phase-ﬁeld Methods for Interfacial Boundaries. Phys. Rev. B: Condens. Matter Mater. Phys., 33(11):7792–7794, 1986.

[29] J. D. Gunton, M. S. Miguel, and P. S. Sahni. The Dynamics of First-Order Phase Transitions. Phase Transitions Crit. Phenom., 8:267–466, 479–82, 1983.

[30] V. L. Ginzburg. Role of Surface Energies in Superconductivity. Physica, 24:S42– S47, 1958.

[31] V. L. Ginzburg. Macroscopic Theory of Superconduction. Soviet Phys., JETP, 2:589–600, 1956.

[32] G. Caginalp and E. A. Socolovsky. Computation of Sharp Phase Boundaries by Spreading : The Planar and Spherically Symmetric Cases. J. Comp. Phys., 95:85–100, 1991.

[33] G. Caginalp. Stefan and Hele-Shaw type Models as Asymptotic Limits of the Phase-ﬁeld Equations. Phys. Rev. A: At., Mol., Opt. Phys., 39:5887–5896, 1989.

177 [34] T. Kyu, R. Mehta, and H. W. Chiu. Spatiotemporal Growth of Faceted and Curved Single Crystals. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 61:4161, 2000.

[35] P. R. Harrowell and D. W. Oxtoby. On the Interaction Between Order and a Moving Interface: Dynamical Disordering and Anisotropic Growth Rates. J. Chem. Phys., 86(5):2932–2942, 1987.

[36] A. A. Wheeler, W. J. Boettinger, and G. B. McFadden. Phase-ﬁeld Model for Isothermal Phase Transitions in Binary Alloys. Phys. Rev. A: At., Mol., Opt. Phys., 45(10):7424–7439, 1992.

[37] A. A. Wheeler, W. J. Boettinger, and G. B. McFadden. Phase-ﬁeld Model of Solute Trapping During Solidiﬁcation. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 47(3):1893–1909, 1993.

[38] A. A. Wheeler, W. J. Boettinger, and G. B. McFadden. Phase-ﬁeld Model of Solute Trapping During Solidiﬁcation. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 47(3):1893–1909, 1993.

[39] H. W. Chiu, Z. L. Zhou, and T. Kyu. Phase Diagrams and Phase Separa- tion Dynamics in Mixtures of Side-Chain Liquid Crystalline Polymers and Low Molar Mass Liquid Crystals. Macromolecules, 29(3):1051–1058, 1996.

[40] H. W. Chiu and T. Kyu. Equilibrium Phase Behavior of Nematic Mixtures. J. Chem. Phys., 103(17):7471–7481, 1995.

[41] H. W. Chiu and T. Kyu. Phase Equilibria of a Nematic and Smectic-A Mixture. J. Chem. Phys., 108(8):3249–3255, 1998.

[42] H. W. Chiu and T. Kyu. Phase Diagrams of a Binary smectic-A Mixture. J. Chem. Phys., 107(17):6859–6866, 1997.

[43] F. Drolet, K. R. Elder, M. Grant, and J. M. Kosterlitz. Phase-ﬁeld Modeling of Eutectic Growth. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 61(6-B):6705–6720, 2000.

[44] K. R. Elder, F. Drolet, J. M. Kosterlitz, and M. Grant. Stochastic Eutectic Growth. Phys. Rev. Lett., 72(5):677–680, 1994.

178 [45] J. C. Wittmann and R. St. J. Manley. Polymer-Monomer Binary Mixtures. Eutectic Crystallization of Poly(ethylene oxide)-Trioxane Mixtures. J. Polym. Sci., Part B: Polym. Phys., 15(12):2277–2280, 1977.

[46] J. C. Wittmann and R. St. J. Manley. Polymer-Monomer Binary Mixtures. I. Eutectic and Epitaxial Crystallization in Poly(e-caprolactone)-Trioxane Mix- tures. J. Polym. Sci., Part B: Polym. Phys., 15(6):1089–1100, 1977.

[47] P. Smith and R. St. J. Manley. Solid Solution Formation and Fractionation in Quasi-Binary Systems of Polyethylene Fractions. Macromolecules, 12(3):483– 491, 1979.

[48] C. H. Klein Douwel, W. Maas, W. S. Veeman, G. H. Werumeus Buning, and J. M. J. Vankan. Miscibility in PMMA/Poly (vinylidene ﬂuoride) Blends, Studied by Fluorine-19-Enhanced Carbon 13 CPMAS NMR. Macromolecules, 23(2):406–412, 1990.

[49] C. Leonard, J. L. Halary, and L. Monnerie. Hydrogen Bonding in PMMA- Fluorinated Polymer Blends: FTi.r. Investigations using Ester Model Molecules. Polymer, 26(10):1507–1513, 1985.

[50] K. J. Kim, Y. J. Cho, and Y. H. Kim. Factors Determining the Formation of the Beta Crystalline Phase of Poly(vinylidene ﬂuoride) in Poly(vinylidene ﬂuoride)-Poly(methyl methacrylate) Blends. Vibrational Spectroscopy, 9:147– 159, 1995.

[51] G. Carini, G. DAngelo, G. Tripodo, A. Bartolotta, G. Di Marco, M. Lanza, V. P. Privalko, and B. Y. Gorodilovand. Locally Heterogeneous Dynamics in Miscible Blends of poly (methyl methacrylate) and poly (vinylidene ﬂuoride). JCP, 116:7316, 2002.

[52] J. Martinez-salazar, J. C. Canalda, and T. Hoﬀmann. On the Melting Behaviour of Polymer Single Crystals in a Mixture with a Compatible Polymer: 1. Poly (vinylidene ﬂuoride)/Poly (methyl methacrylate) Blends. Polymer, 36(5):981, 1995.

[53] H. Tanaka and T. Nishi. Local Phase Separation at the Growth Front of a Polymer Spherulite During Crystallization and Nonlinear Spherulitic Growth in a Polymer Mixture with a Phase Diagram. Phys. Rev. A: At., Mol., Opt. Phys., 39(2):783–794, 1989.

179 [54] C. Y. Chen, W. Yunus, and T. Kyu. Phase Separation Behaviour in Blends of Isotactic Polypropylene and Ethylene-Propylene Diene Terpolymer. Polymer, 38(17):4433–4438, 1997.

[55] A. Ramanujam, K. J. Kim, and T. Kyu. Phase Diagram, Morphology Devel- opment and Vulcanization Induced Phase Separation in Blends of Syndiotactic Polypropylene and Ethylene-Propylene Diene Terpolymer. Polymer, 41(14):5375– 5383, 2000.

[56] D.A. Kessler, J. Koplik, and H. Levine. Pattern Selection in Fingered Growth Phenomena. Advances in Physics, 37(3):255–339, 1988.

[57] T. Kyu, H. Xu, and R. Matkar. Phase-ﬁeld Modeling on Morphological Land- scape of Isotactic Polystyrene Single Crystals. Phys. Rev. E: Stat. Phys., Plas- mas, Fluids, Relat. Interdiscip. Top., 72(1):11804, 2005.

[58] J. W. Cahn. Later Stages of Spinodal Decomposition and the Beginnings of Particle Coarsening. Acta Metallurgica, 14(12):1685–1692, 1966.

[59] J. W. Cahn. Phase Separation by Spinodal Decomposition in Isotropic Systems. J. Chem. Phys., 42(1):93–99, 1965.

[60] De Gennes P. G. Dynamics of Fluctuations and Spinodal Decomposition in Polymer Blends. J. Chem. Phys., 72:4756, 1980.

[61] K. Binder. Collective Diﬀusion, Nucleation, and Spinodal Decomposition in Polymer Mixtures. J. Chem. Phys., 79:6387, 1983.

[62] J. W. Cahn. Free Energy of a Nonuniform System. II. Thermodynamic Basis. J. Chem. Phys., 30:1121, 1959.

[63] J. W. Cahn and J. E. Hillard. Free Energy of a Nonuniform System. III. Nucleation in a Two-Component Incompressible Fluid. J. Chem. Phys., 31:688, 1959.

[64] J. W. Cahn. Phase Separation by Spinodal Decomposition in Isotropic Systems. J. Chem. Phys., 42:93, 1965.

[65] N. C. Billingham and P. D. Calvert. Studies of Diﬀusion in Polymers by Ultra- violet Microscopy: 2. Diﬀusion of Atactic Fractions in Isotactic Polypropylene. Polymer, 31:258–264, 1990.

180 [66] P. J. Flory. Thermodynamics of Crystallization in High Polymers. IV. A Theory of Crystalline States and Fusion in Polymers, Copolymers, and Their Mixtures with Diluents. J. Chem. Phys., 17:223–240, 1949.

[67] R. Koningsveld and W. H. Stockmayer. Polymer Phase Diagrams. Oxford University Press, N.Y., 2001.

[68] S. Malik, C. Rochas, and J. M. Guenet. Syndiotactic Polystyrene/Naphthalene Intercalates: Preparing Thermoreversible Fibrillar Gels from a Solid Solvent. Macromolecules, 38(11):4888–4893, 2005.

[69] S. Malik, C. Rochas, M. Schmutz, and J. M. Guenet. Syndiotactic Polystyrene Intercalates from Naphthalene Derivatives. Macromolecules, 38(14):6024–6030, 2005.

[70] S. Malik, C. Rochas, and J. M. Guenet. Thermodynamic and Structural In- vestigations on the Diﬀerent Forms of Syndiotactic Polystyrene Intercalates. Macromolecules, 39(3):1000–1007, 2006.

[71] T. Kyu and H. W. Chiu. Phase Equilibria of a Polymer–Smectic-Liquid-Crystal Mixture. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 53(4):3618–3622, 1996.

[72] B. Hahn, J. Wendorﬀ, and D. Y. Yoon. Dielectric Relaxation of the Crystal- Amorphous Interphase in Poly (vinylidene ﬂuoride) and its Blends with Poly (methyl methacrylate). Macromolecules, 18(4):718–721, 1985.

[73] R. S. Stein and G. Hadziioannou. Neutron Scattering Studies of Dimensions and of Interactions Between Components in Polystyrene/Poly (vinyl methyl ether) and Poly (vinylidene ﬂuoride)/Poly (methyl methacrylate) Amorphous Blends. Macromolecules, 17:567–573, 1984.

[74] J. H. Wendorﬀ. Concentration Fluctuations in Poly (vinylidene ﬂuoride)-Poly (methyl methacrylate) mixtures. J. Polym. Sci., Polym. Lett. Ed., 18:439, 1980.

[75] S. M. Allen and J. W. Cahn. Mechanisms of Phase Transformations Within the Miscibility Gap of Fe-rich Fe-A1 Alloys. Acta Metall., 24:425–437, 1976.

[76] R. Kerridge. Melting-point Diagrams for Binary Triglyceride Systems. J. Chem. Soc., Abstracts, pages 4577–4579, 1952.

181 [77] P. Smith and J. Pennings. Eutectic Solidiﬁcation of the Quasi Binary System of Isotactic Polypropylene and Pentaerythrityl Tetrabromide. J. Polym. Sci., Part B: Polym. Phys., 15:523–540, 1977.

[78] S. K. Chan. Steady-State Kinetics of Diﬀusionless First Order Phase Transfor- mations. J. Chem. Phys., 67:5755–5762, 1977.

[79] C. Shen and T. Kyu. Spinodals in a Polymer Dispersed Liquid Crystal. J. Chem. Phys., 102(1):556–562, 1998.

[80] S. M. Allen and J. W. Cahn. Mechanisms of Phase Transformations within the Miscibility Gap of Fe-Rich Fe-A1 Alloys. Acta Metall., 24:425–437, 1976.

[81] G. Caginalp and P. Fife. Phase-Field Methods for Interfacial Boundaries. Phys. Rev. B: Condens. Matter Mater. Phys., 33(11):7792–7794, 1986.

[82] A. A. Wheeler, W. J. Boettinger, and G. B. McFadden. Phase-ﬁeld Model for Isothermal Phase Transitions in Binary Alloys. Phys. Rev. A: At., Mol., Opt. Phys., 45(10):7424–7439, 1992.

[83] R. Kobayashi. Modeling and Numerical Simulations of Dendritic Crystal Growth. Physica D, 63:410–423, 1993.

[84] A. Karma and W. Rappel. Quantitative Phase-ﬁeld Modeling of Dendritic Growth in Two and Three Dimensions. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 57:4323–4349, 1998.

[85] K. R. Elder, J. D. Gunton, and M. Grant. Nonisothermal Eutectic Crystal- lization. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 54:6476, 1996.

[86] P. Mathis and A. Karma. Eutectic Colony Formation: A Stability Analysis. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 60:6865, 1999.

[87] R. A. Matkar and T. Kyu. Phase diagrams of binary crystalline-crystalline polymer blends. J. Phys. Chem. B, 110(32):16059–16065, 2006.

[88] H. D. Keith and F. J. Padden. A phenomenological Theory of Spherulitic Crystallization. J. Appl. Phys., 34:2409–2421, 1963.

182 [89] H. D. Keith and F. J. Padden. Spherulitic Crystallization from the Melt. I. Fractionation and Impurity Segregation and Their Inﬂuence on Crystalline Mor- phology. J. Appl. Phys., 35:1270–1285, 1964.

[90] N. C. Billingham, P. D. Calvert, and A. Uzuner. Diﬀusion of Atactic Fractions in Isotactic Polypropylene. Polymer, 31:258, 1990.

[91] D. J. Lohse. The Melt Compatibility of Blends of Polypropylene and Ethylene- Propylene Copolymers. Poly. Eng. and Sci., 26:1500–1509, 2004.

[92] Z. Wang, R. A. Phillips, and B. S. Hsiao. Morphology Development During Isothermal Crystallization. I. Isotactic and Atactic Polypropylene Blends. J. Polym. Sci., Part B: Polym. Phys., 38:2580–2590, 2000.

[93] A. K. Bhowmick and H. L. Stephens. Handbook of Elastomers. Marcel Dekker Inc, 2000.

[94] L. A. Goettler, J. R. Richwine, and F. J. Wille. The rheology and processing of oleﬁn-based thermoplastic vulcanizates. Rubber Chem. Technol., 55(5):1, 1982.

[95] A. Y. Coran and R. P. Patel. Rubber-thermoplastic compositions part i. epdm- polypropylene thermoplastic vulcanizates. Rubber Chem. Technol, 53(1):141– 115, 1980.

[96] A. Y. Coran and R. P. Patel. Rubber-thermoplastic compositions. part iv. thermoplastic vulcanizates from various rubber-plastic combinations. Rubber Chem. Technol., 54:892–903, 1981.

[97] T. Kyu and H. W. Chiu. Phase Equilibria of a Polymer–Smectic-Liquid-Crystal Mixture. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 53:3618, 1996.

[98] J. Brandrup and E. H. Immergut. Polymer Handbook. Wiley New York, 1975.

[99] Y. W. Cheung and R. S. Stein. Critical Analysis of the Phase Behavior of Poly (-caprolactone)(PCL)/Polycarbonate (PC) Blends. Macromolecules, 27(9):2512– 2519, 1994.

[100] Y. W. Cheung, J. S. Lin, and R. S. Stein. Small-Angle Scattering Investi- gations of Poly (-caprolactone)/Polycarbonate Blends. 2. Small-Angle X-ray and Light Scattering Study of Semicrystalline/Semicrystalline and Semicrys- talline/Amorphous Blend Morphologies. Macromolecules, 27(9):2520–2528, 1994.

183 [101] P. M. Cham, T. H. Lee, and H. Marand. On the State of Miscibility of Isotactic Poly (propylene)/Isotactic Poly (1-butene) Blends: Competitive Liquid-Liquid Demixing and Crystallization Processes. Macromolecules, 27(15):4263–4273, 1994.

[102] J. P. Penning and R. St. J. Manley. Miscible Blends of Two Crystalline Poly- mers. 1. Phase Behavior and Miscibility in Blends of Poly (vinylidene ﬂuoride) and Poly (1, 4-butylene adipate). Macromolecules, 29(1):77–83, 1996.

[103] J. P. Penning and R. St. J. Manley. Miscible Blends of Two Crystalline Poly- mers. 2. Crystallization Kinetics and Morphology in Blends of Poly (vinylidene ﬂuoride) and Poly (1, 4-butylene adipate). Macromolecules, 29:84–90, 1996.

[104] K. Fujita, T. Kyu, and R. St. J. Manley. Miscible Blends of Two Crystalline Polymers. 3. Liquid-Liquid Phase Separation in Blends of Poly(vinylidene ﬂu- oride)/Poly(butylene adipate). Macromolecules, 29(1):91–96, 1996.

[105] N. Inaba, T. Yamada, and T. Hashimoto. Morphology Control of Binary Poly- mer Mixtures by Spinodal Decomposition and Crystallization. 2. Further Stud- ies on PP/EPR. Macromolecules, 21(2):407–414, 1988.

[106] B. Crist and M. J. Hill. Recent Developments in Phase Separation of Polyoleﬁn Melt Blends. J. Polym. Sci., Part B: Polym. Phys., 35(14):2329–2353, 1997.

184 APPENDICES

185 APPENDIX A

COMMON TANGENT METHOD

The requirement of a continuous function in φ is a drawback of the Michelson method of minimization. In the present thesis, I have modiﬁed this approach and developed an algorithm that can be coded to run very eﬃciently on vector machines and has been included below. The condition for balancing chemical potentials and minimization of the free energy can be analyzed from a geometrical perspective by analyzing the free energy curve as a function of the single independent variable φ which yield the common tangent criterion. There exist two points on the free energy curve that have the same slope on the free energy curve, which is also equal to the slope of the line joining these two points. This criterion determines the coexistence compositions φα and φβ according to the relation

α β ∂∆Gm ∂∆Gm ∆Gm − ∆Gm = = α β (A.1) ∂φ φα ∂φ φβ φ − φ

The algorithm attacks the problem of ﬁnding cotangents from a geometrical perspective. Tangents to the curve can be classiﬁed into 2 categories.

1. Non-intersecting tangents: Tangent lying completely below the free energy function.

186 2. Intersecting tangents: Tangent which intersects the free energy function such that some portion of the tangent lie above the free energy curve. The phase boundaries are the locus of all the points that are common tangents to the free energy function.

The common tangents are deﬁned as the tangents to the free energy function that are tangents to at least two unique values of x. The locus of common tangents is equivalent to the locus of all points at which the transition of the intersection property of the tangents takes place. This can be expressed as a step function p(x) such that

p(x) = 1; Intersecting (A.2) 0; Non-intersecting

For establishing polymer-solvent phase diagrams that usually have highly skewed free energy wells, the Michelson method fails whereas the algorithm proposed, is strikingly successful as it only approximates the cotangent points to the resolution of the free energy function. Multiple cotangents, tricritical points can also be estimated using this approach.

187 Algorithm 1 Phase Stability Algorithm for i = 1 to N do Calculate f(φi) if Gradient information available then ∂f Calculate ∂φ i else ∂f f(φi+1) − f(φi−1) Calculate = ∂φ i 2/N end if end for A N 2 loop to calculate line equations of each tangent in the range 0 − 1 for i = 2 to N − 1 do Tangent line equation

∂∆f m = ∂φ i Y int = f(φi) − m(i)φi for j = 2 to N − 1 do yj = mφj + Y int end for Compare functions yj and fj to determine pj if pj changes from 0 to 1 or 1 to 0 then φi is one of the phase boundaries end if end for

188 550 550 550

500 500 500 T(K) T (K) T (K) 450 450 450

400 400 400 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 0 0.1 0.2 0.3 0.4 0.5 φ φ φ (a) (b) (c)

Figure A.1: Demonstration of the modiﬁed common tangent algorithm to accurately predict phase diagrams even for highly asymmetric mixtures. X1/X2 = 1, 10, 100 from left to right

189 APPENDIX B

EMPIRICAL RELATIONS OF χ

χaa = B/T (B.1)

χaa = A + B/T (B.2)

χaa = A + B/T + Cln(T ) (B.3)

We have used three forms of χaa (Equations (B.1, B.2 and B.3) during the course of this dissertation out of which the last two are empirical. If the statistical segment lengths of the polymers (r1 and r2) are known and if the upper critical solution temperature (UCST) or lower critical solution temperature (LCST) or both are known, it is possible to reduce the last two equations to be a function of A, TUCST and TLCST alone. The ﬁrst equation can only predict a UCST curve as χaa can only decrease with increasing temperature. As has already been shown in Chapter 2, at T = Tcrit,

χ = χcrit where √ √ 2 ( r1 + r2) χcrit = (B.4) 2r1r2

190 Here Tcrit is the UCST. Thus we can ﬁnd B,

B = χcritTcrit (B.5)

On substitution of B in Equation (B.1), we get

χaa = χcritTcrit/T (B.6)

This form of the relation for χaa requires only information about the Tcrit. Similarly, we can apply the same procedure to Equation (B.2) yielding

B = (χcrit − A)Tcrit (B.7)

On substitution of B in Equation (B.1), we get

χaa = A + (χcrit − A)Tcrit/T (B.8)

In this relation, the magnitude of A determines whether the predicted phase diagram will show a UCST or LCST behavior. If A > χcrit, B becomes negative and we get an

LCST behavior and if A < χcrit, B is always positive and we get a UCST behavior.

If the binary mixture exhibits both UCST and LCST behaviors, then we get two relations, at T = TUCST , χ = χcrit and at T = TLCST , χ = χcrit. These can be

191 plugged into Equation (B.3) to get two equations.

χcrit = A + B/TUCST + Cln(TUCST ) (B.9a)

χcrit = A + B/TLCST + Cln(TLCST ) (B.9b)

By multiplying Equation (B.9a) with TUCST and Equation (B.9b) with TLCST and subtracting one from the other, we can eliminate B yielding

(χcrit − A)(TUCST − TLCST ) = C(TUCST ln(TUCST ) − TLCST ln(TLCST )) (B.10)

Thus we get the relation for C,

(χ − A)(T − T ) C = crit UCST LCST (B.11) (TUCST ln(TUCST ) − TLCST ln(TLCST )

By dividing Equation (B.9a) by ln(TUCST ) and Equation (B.9b) by ln(TLCST ) and subtracting one from the other, we can eliminate C yielding

1 1 1 1 (χcrit − A)( − ) = B( − ) ln(TUCST ) ln(TLCST ) TUCST ln(TUCST ) TLCST ln(TLCST ) (B.12)

192 Thus, we get the relation for B as follows

1 1 (χ − A)( − ) crit ln(T ) ln(T ) B = UCST LCST (B.13) 1 1 ( − ) TUCST ln(TUCST ) TLCST ln(TLCST )

This form of the equation requires three unknowns A, TLCST and TUCST . If A > χcrit, we get a hour glass phase diagram; while if A < χcrit, we get a close loop phase diagram. The use of either of these equations enables us to predict many kinds of liquid-liquid phase separation behavior.

193 APPENDIX C

CONVERSION OF MATERIAL TO MODEL PARAMETERS

In the calculation of the phase diagrams, we have several model parameters that are correlated to material parameters of the blend. In this dissertation, the free energy of crystallization serves as the starting point to determine the model parameters. The model parameter ζ0 has been taken as 1 for all the calculations. This allows us to use Equation (2.42) that serves as the starting point in the calculation of the rest of the model parameters from material parameters.

∆Hu T ζ 1 ∆F = 1 − 0 = W − (C.1) RT Tm 6 12

In this relation, we know from Equation (2.49) that

1 ∆Huδ T ζ = − (1 − 0 ) (C.2) 2 Rσ Tm

This relation requires the knowledge of the surface energy of the crystal (σ) and the phase ﬁeld interface thickness (δ) which are usually hard to get. Kobayashi [83] has

194 suggested a particular form of ζ that mimics the behavior of Equation (C.2),

0 ζ = 0.5 − 0.9 arctan((1 − T/Tm))/π (C.3)

0 This is a sigmoidal function that is centered (T = Tm) around 0.5. Using this value of ζ we calculate W using the relation

∆Hu T ζ 1 W = 1 − 0 / − (C.4) RT Tm 6 12

The procedure to calculate the crystalline-amorphous interaction energy and the amorphous-amorphous interaction energy has already been described in Chapter 3.

195