12728
J. Phys. Chem. B 2006, 110, 12728-12732
Role of Crystal-Amorphous Interaction in Phase Equilibria of Crystal-Amorphous Polymer Blends Rushikesh A. Matkar and Thein Kyu* Department of Polymer Engineering, UniVersity of Akron, Akron, Ohio 44325-0301 ReceiVed: February 23, 2006; In Final Form: May 3, 2006
A self-consistent theory has been developed for determination of phase diagrams of a crystalline polymer solution. Although the original Flory diluent theory captures the liquidus line, the theory is incapable of accounting for the solidus line due to the inherent assumption of complete immiscibility of solvent in the solid crystal. The present theory considers all possible interactions involving amorphous-amorphous and crystal-amorphous interactions. The self-consistent solutions predict various phase diagrams involving liquidliquid, pure solid, and liquid-solid coexistence regions bound by liquidus and solidus lines. In the limit of complete insolubility of solvent in neat solid crystal, the original Flory diluent theory is recovered.
Introduction More than half a century ago, the solubility of a polymer solute in a monomeric solvent was theoretically deduced by Flory by relaxing two assumptions inherent in the regular solution theory;1 viz., (i) the constituent polymer solvent and solvent are dissimilar and (ii) the mixture is nonideal, 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.2 The Flory diluent theory for polymer solutions was later extended by Nishi and Wang3 to determine the Flory-Huggins interaction parameter, χ, from the melting point depression data of crystalline (polyvinylidene difluoride, PVDF)/amorphous (polymethyl methacrylate, PMMA) polymer blends under the aforementioned assumption of complete immiscibility1 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 small-angle neutron and X-ray scattering.4 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, and thus it may be attributed unambiguously to the segmental amorphousamorphous 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 amorphousamorphous interaction or the amorphous-crystalline chain interaction or the combination of both. This ambiguity went on unchecked despite the reported discrepancy between the χ * Corresponding author. E-mail:
[email protected].
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.5 In particular, the Flory theory may be valid strictly for completely miscible 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. This shortcoming implies a potential flaw such as multiplicity of roots also known as the “van der Waals” loop artifact.6 A natural question is what happens to the phase diagrams if this complete insolubility assumption is removed. In the present work, we relax the last assumption of complete immiscibility of the solvent in the crystal by incorporating a crystal-solvent interaction parameter in addition to the amorphous-amorphous interaction parameter. A variety of phase diagrams for a crystalline-amorphous polymer blend have been constructed by incorporating the amorphous-amorphous and crystal-amorphous interactions of the constituent polymers in the free-energy description. Various coexistence curves have been self-consistently 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. Free-Energy Landscape of a Crystalline Polymer Blend. We shall consider a crystalline polymer blend in which only one component can crystallize and the other is a noncrystallizable amorphous polymer. The total free-energy density of mixing of a crystal-amorphous polymer blend may be expressed as the weighted sum of the free-energy density pertaining to crystal order parameter7 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.8 The 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
10.1021/jp061159m CCC: $33.50 © 2006 American Chemical Society Published on Web 06/03/2006
Crystal-Amorphous Interaction in Polymer Blends
J. Phys. Chem. B, Vol. 110, No. 25, 2006 12729
Figure 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 eq 1 (i.e., ∂f/∂ψ ) 0).
retained in the Flory diluent theory by taking into consideration all possible interaction terms such as amorphous-amorphous and crystal-amorphous interactions, viz.,
(1 - φ) φ f(ψ,φ) ) φf(ψ) + ln(φ) + ln(1 - φ) + r1 r2 {χaa + χcaψ2}φ(1 - φ) (1) where χaa is the Flory-Huggins interaction parameter representing the amorphous-amorphous interaction of the constituent chains in the isotropic melt. χca represents the crystalamorphous interaction parameter. Note that the order of the subscripts denotes the constituent 1 (crystal) and constituent 2 (amorphous polymer). r1 and r2 correspond to the statistical segmental lengths of the respective components. To clarify the physical essence of the enthalpic contribution, eq 1 may be rewritten as
(1 - φ) φ ln(1 - φ) + f(ψ,φ) ) φf(ψ) + ln(φ) + r1 r2 χaaφ(1 - φ) + χca[φψ][(1 - φ)ψ] (2) 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), that is, 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 eq 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 crystal-amorphous interaction. The aforementioned crystal order parameter (ψ) may be described in terms of the Landau-type free-energy expansion, viz.,
f(ψ) )
[
]
ζ(T)ζ0(Tm) 2 ζ(T) + ζ0(Tm) 3 1 4 F(ψ) )W ψ ψ + ψ kBT 2 3 4 (3)
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 phenomenon such as crystallization.7 It should be cautioned that the coefficient of the cubic power term must be nonzero to apply the Landau potential to the first-order phase transition. Otherwise, eq 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.7 The uniqueness of the present free-energy description is that the crystal order parameter at the solidification potential strongly depends on the supercooling, 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., nonequilibrium) 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 supercoolings. This type of asymmetric Landau potential has been employed successfully to elucidating the dynamics of crystallization and morphology evolution of neat syndiotactic polystyrene showing the spatiotemporal emergence of hierarchy structures encompassing faceted hexagonal single crystals, snowflake, seaweed, and dense lamellar morphology (i.e., spherulites).7 In the present paper, we shall focus only on equilibrium aspects of thermodynamic phase diagrams of crystal-amorphous polymer blends. Completely Miscible Systems. Figure 1a shows the freeenergy 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 in determining the roots at ∂f/∂ψ ) 0 (Figure 1a, solid dots). Subsequently, the chemical potentials were calculated through minimization with respect to volume fraction (φ), and the coexistence loci were determined with the aid of the common tangent algorithm.9 As usual, both interaction parameters are taken to be inversely proportional to absolute temperature.
12730 J. Phys. Chem. B, Vol. 110, No. 25, 2006
Matkar and Kyu
Figure 2. Effect of increasing the crystal-amorphous interaction parameter χca ) (a) 0.01, (b) 0.1, and (c) 0.3 at 500 K. (d) Effect of increasing the amorphous-amorphous interaction energy for a finite value of crystal-amorphous interaction parameter χca ) 0.3 at 500 K. The liquidus curve shifts downward as the intermolecular interaction of the liquid becomes more attractive and shifts upward as the intermolecular interaction of the liquid becomes less attractive.
For illustration purposes, a hypothetical polymer phase diagram was first calculated showing an overlap of the solidus and liquidus lines (Figure 2a), assuming heat of crystallization to be 3 kcal/mol, r1 ) 100, r2 ) 30, and χca ) 0.01 at the melting temperature (500 K) of the neat crystallizable component. By increasing the repulsive crystal-amorphous interaction parameter to χca ) 0.1, the solidus and liquidus lines get separated, showing the coexistence of the crystal + liquid (amorphous) region (Figure 2b). As χca increases, the solidus and liquidus curves both shift downward, but the crystalamorphous coexistence gap broadens, implying that the solidus line moves down much faster than the liquidus curve. At the crystal-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 2c). This trend remains virtually unchanged with further increase of the repulsive crystal-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 crystal-amorphous blends. Figure 2d demonstrates the effect of the attractive amorphousamorphous interaction parameter (i.e., negative χaa) on the liquidus line at the limiting value of the crystal-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 point depression that represents the χaa amorphous-amorphous interaction at a given crystal-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 in comparison with the melting point depression data of the crystalline-amorphous blend of PVDF and PMMA reported by Nishi and Wang.3 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. From the theoretical fit with the experimental melting 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 weightdependent as in the case of the FH interaction parameter, χaa. The estimated value of χaa ) -0.141 is smaller in magnitude (i.e., lesser attractive interaction) than the reported result by Nishi and Wang; that is, 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, for example, χaa ≈ -0.226 and -0.077 for two different concentrations of PVDF/PMMA by Stein and Hadziioannou10 and -0.14 by Canalda et al.,4 and -0.03/-0.16 from SAXS by Wendorff.11 In the Flory diluent theory, the repulsive crystal-amorphous interaction χca was completely ignored, and thus the χaa had
Crystal-Amorphous Interaction in Polymer Blends
J. Phys. Chem. B, Vol. 110, No. 25, 2006 12731 solidus line that has been overlooked in the determination of polymer phase diagrams due to the poor assumption of the complete insolubility of the solvent in the polymer solute. In practice, the solidus line is hard to obtain experimentally due to the nonequilibrium nature of polymer crystallization and the uncertainty of the broad DSC peaks.13 Occasionally, even if two distinct DSC peaks were observed experimentally, the interpretation has been biased toward 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 the 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.
Figure 3. Theoretically calculated phase diagram of PVDF/PMMA blend (b). Experimental data and material parameters were determined by Nishi and Wang (]).
been overestimated as compared to those values from 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 may be obtained from the melting point depression data. This kind of uncertainty may be eliminated completely in a partially miscible system where the solid-liquid-phase transition is competing with the UCST type liquid-liquid-phase separation. Partially Miscible Systems. In the case of nonideal mixing, liquid-liquid-phase separation is expected to occur in competition with the melting transition of the crystallizable component.12 Figure 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 (450 K) below the melting transition (500 K) of the crystalline constituent. With increasing contribution of the liquid-liquid demixing, the critical temperature (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, liquidliquid, 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
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.14 The experimental data were replotted in Figure 5a 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 33 000 for PCL). The UCST peak is located 510 K higher than the melting temperatures of PCL (333 K) in the blends. χaa was estimated directly from the UCST (i.e., χaa ) 1.04 at 333 K). Using this χaa value, we constructed the melting point depression curve according to the original Flory diluent theory, which ended predicting a much suppressed liquidus trend (the crosses in Figure 5b), indicating the failure of the Flory diluent theory for this partially miscible PCL/PS blend. As advocated by the present theory, the crystal-amorphous interaction parameter, χca, and the amorphous-amorphous interaction parameter, χaa, contribute to the UCST as well as to 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 crystal-amorphous interaction parameter is evaluated to be χca ) 0.46 at 333 K 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, χaa, can be calculated directly from the UCST part of the phase diagram, whereas the χca crystal-amorphous interaction parameter can be determined from the crystal-melting transition.
Figure 4. Predicted phase diagrams for crystal-amorphous mixtures exhibiting (a) isotropic, crystal-liquid, neat crystal region, and the melting transition of the crystallizable component and (b) a teapot-type phase diagram showing the liquid-liquid coexistence region as the upper critical solution temperature envelope protruded above the liquidus line of Figure 4a.
12732 J. Phys. Chem. B, Vol. 110, No. 25, 2006
Matkar and Kyu
Figure 5. (a) Comparison between the experimental phase diagram of PCL/PS blend by Tanaka and Nishi (]) and the self-consistent calculation represented by the solid line. (b) 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 (×) calculated by the original Flory diluent theory using the value of χaa ) 1.04.
Conclusions We have developed a new self-consistent theory for the determination of phase diagrams of a crystalline polymer solution. The modification was made to the original Flory diluent theory; that is, we relaxed the complete immiscibility assumption of the polymeric solvent in the neat solid crystal by taking into consideration the crystal-amorphous interaction in addition to the amorphous-amorphous interaction of the pair. The present paper is the first to point out the significance of the crystalamorphous interaction in the determination of the phase diagrams of crystalline polymer solutions. Ignoring the crystalamorphous interaction χca, the original Flory diluent theory results in overpredicting the χaa value. 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 crystal-amorphous polymer blends capture both solidus and liquidus lines forming eutectics and peritectic 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.
Acknowledgment. Support from the National Science Foundation (NSF) through Grant No. DMR-0514942 is gratefully acknowledged. References and Notes (1) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice-Hall: New York, 1999. (2) Flory, P. J. J. Chem. Phys. 1949, 17 (3), 223. (3) Nishi, T.; Wang, T. T. Macromolecules 1975, 8 (6), 909. (4) Canalda, J. C.; Hoffmann, Th.; Martinez-Salazar, J. Polymer 1995, 36 (5), 981. (5) Konigsveld, R.; Stockmayer, W. H. Polymer Phase Diagrams; Oxford University Press: Oxford, New York, 2001. (6) Burghardt, W. R. Macromolecules 1989, 22 (5), 2482. (7) Xu, H.; Matkar, R.; Kyu, T. Phys. ReV. E 2005, 72, 011804. (8) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (9) Kyu, T.; Chiu, H. W. Phys. ReV. E 1996, 53, 3618. (10) Hadziioannou, G.; Stein, R. S. Macromolecules 1984, 17 (4), 567. (11) Wendorff, J. H. J. Polym. Sci., Polym. Lett. Ed. 1980, 18, 439. (12) Allen, M. S.; Cahn, J. W. Scr. Metall. 1976, 10, 451. (13) Smith, P.; Manley, R. St. J. Macromolecules 1979, 12 (3), 483. (14) Tanaka, H.; Nishi, T. Phys. ReV. A 1989, 39 (2), 783.
