J. Phys. Chem. B 2006, 110, 16059-16065
16059
Phase Diagrams of Binary Crystalline-Crystalline Polymer Blends Rushikesh A. Matkar and Thein Kyu* Department of Polymer Engineering, UniVersity of Akron, Akron, Ohio 44325 ReceiVed: April 5, 2006; In Final Form: June 27, 2006
A thermodynamically self-consistent theory has been developed to establish binary phase diagrams for twocrystalline polymer blends by taking into consideration all interactions including amorphous-amorphous, crystal-amorphous, amorphous-crystal, and crystal-crystal interactions. The present theory basically involves combination of the Flory-Huggins free energy for amorphous-amorphous isotropic mixing and the Landau free energy of polymer solidification (e.g., crystallization) of the crystalline constituents. The self-consistent solution via minimization of the free energy of the mixture affords determination of eutectic, peritectic, and azeotrope phase diagrams involving various coexistence regions such as liquid-liquid, liquid-solid, and solid-solid coexistence regions bound by liquidus and solidus lines. To validate the present theory, the predicted eutectic phase diagrams have been compared with the reported experimental binary phase diagrams of blends such as polyethylene fractions as well as polycaprolactone/trioxane mixtures.
Introduction Eutectic crystallization has been extensively explored in metal alloys and small molecule systems,1,2 but such a phenomenon has received little attention in binary crystalline polymer blends,3 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.4 Subsequently, Manley et al. reported the eutectic crystallization of poly(caprolactone) in trioxane as well as in blends of polyethylene fractions.5,6 The observed eutectic curves were analyzed in the framework of the Flory diluent theory7 by individually treating the melting point depression of the constituent crystals to be 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 that is incapable of explaining the solidus line in the phase diagram because of the inherent immiscibility assumption 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 major goal of the present paper is to elucidate the phenomenon of eutectic, peritectic, or azeotropic crystallization in polymer blends by taking into consideration the solid solution phase that has been known to exist in other binary systems such as metal alloys, organic molecular solutions, and liquid crystalline mixtures. A unified 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 transition8-13 coupled with the Flory-Huggins free energy for liquid-liquid demixing.14-16 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 * Address correspondence to this author.
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 well with the experimental phase diagrams reported in the literature. 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 1), viz.
f(ψ) )
F(ψ) )W kBT
)W
∫0ψ ψ(ψ - ζ)(ψ - ζ0) dψ
[
]
ζ(T)ζ0(Tm) 2 ζ(T) + ζ0(Tm) 3 1 4 ψ ψ + ψ 2 3 4
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., ψ ) l/l0, and thus it represents the one-dimensional crystallinity, hereafter called linear crystallinity.12,13 This kind of asymmetric Landau potential has been utilized in the phase field model to explain the dynamics of solidification phenomenon such as crystal growth. It should be cautioned that the coefficient of the cubic term must be nonzero in order to apply the aforementioned Landau potential to the first-order phase transition; otherwise, eq 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
10.1021/jp062124p CCC: $33.50 © 2006 American Chemical Society Published on Web 07/27/2006
16060 J. Phys. Chem. B, Vol. 110, No. 32, 2006
Matkar and Kyu crystalline constituent with its volume fraction (φ) and the free energy of liquid-liquid mixing as described by the FloryHuggin’s theory of isotropic mixing14-16 with the addition of the anisotropic interaction terms including crystal-amorphous, amorphous-crystal, and crystal-crystal mixing in what follows:
f (ψ1,ψ2,φ) ) φ f (ψ1) + (1 - φ) f (ψ2) +
φ ln(φ) + r1
(1 - φ) ln(1 - φ) + {χaa + (χcaψ12 - 2χccψ1ψ2 + r2 χacψ22)}φ(1 - φ) (2)
Figure 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 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.
represents the penalty for the nucleation process. ζ0 represents the crystal order parameter at the solidification potential for crystallization that may be treated as supercooling or crystal melting temperature dependent (Figure 1). In principle, the stable solid may vary from unity at equilibrium to some finite values of ζ0 depending on the supercooling or the melting temperature. At Tm0, the free energy densities of the melt and the solid are equivalent, implying the coexistence of the crystal and melt. When T < Tm0, 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 ψ axis. As the supercooling increases, ζ0 moves to a lower value, which implies the imperfect crystal containing a 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.12,13 Such nonequilibrium growth dynamics will not be dealt with here as it is beyond the scope of the present article. 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
The first term represents the Landau-type free energy of crystal solidification of each component in which the individual free energy of the constituents is weighted by the respective volume fractions to ensure that these potentials vanish at the extreme limits of zero crystallinity or if a component is noncrystallizable. The second term represents the entropic part of the free energy of mixing of the amorphous constituents in which r1 and r2 are the statistical segment lengths of the constituent polymer chains. The third 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 crystal solid-amorphous liquid or amorphous liquid-crystal solid interactions may be defined as χca and χac, respectively. These anisotropic interactions of separate crystals and χcc of cocrystals 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 enthalpies of crystallization, i.e.,
χca ∼ ∆H1c/RT and χac ∼ ∆H2c/RT
(3)
where ∆H1c and ∆H2c are enthalpies of crystallization of components 1 and 2, respectively. Furthermore, the crystalcrystal interaction may be expressed as a geometric mean of the crystal-amorphous and amorphous-crystal interactions to account for the nonideal rule of the crystalline mixture, i.e., χc1c2 ) cω12(xχc1a2‚xχa1c2) or χcc ) cω(xχca‚xχac) 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 fusion of these individual crystals as well as of the cocrystals and thus it is not directly connected to χaa. This cω parameter is analogous to the “c” anisotropic interaction parameter defined for the cases of smectic and nematic liquid crystal mixtures.17,18 Physically, the terms in the small bracket of eq 2 are explicable in terms of the crystal-amorphous, crystal-crystal, and amorphous-crystal interactions. 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 crystal-amorphous interaction. The same argument may be made to the second crystalline component, i.e., χacψ2(1 - φ)ψ2φ represents the amorphouscrystal interaction. On the same token, the cross-interaction term, χccψ1φψ2(1 - φ), may be interpreted as the crystal-crystal interaction.
Binary Crystalline-Crystalline Polymer Blends
J. Phys. Chem. B, Vol. 110, No. 32, 2006 16061
Construction of Phase Diagrams via Free Energy Minimization To construct the phase transition points and determine the coexistence curves, we first minimize the free energy with respect to all nonconserved 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 Hessian matrix. The Gauss-Newton method of minimization is a recursive algorithm to minimize a multivariable function
∂ f (ψ1,ψ2,φ) ∂ f (ψ1,ψ2,φ) | φR ) |φβ ) ∂φ ∂φ f (ψ1,ψ2,φR) - f (ψ1,ψ2,φβ) φR - φβ
(7)
The detailed description of the aforementioned approach may be found elsewhere.17,18 Results and Discussion
ψin+1 ) ψin - [H( f)]-1∇f
(4)
where the gradient vector and the Hessian matrix are defined as
[ ]
∂ f (ψ1,ψ2,φ) ∂ψ1 ∇f ) ∂ f (ψ1,ψ2,φ) ∂ψ2 and
H( f ) )
[
∂2f (ψ1,ψ2,φ) ∂2f (ψ1,ψ2,φ) ∂ψ1∂ψ2 ∂ψ12
2 ∂2f (ψ1,ψ2,φ) ∂ f (ψ1,ψ2,φ) ∂ψ2∂ψ1 ∂ψ22
]
(5)
where
∂ f (ψ1,ψ2,φ) ) φ f ′(ψ1) + φ(1 - φ)(2χcaψ1 - 2χccψ2) ∂ψ1 ∂ f (ψ1,ψ2,φ) ) (1 - φ) f ′(ψ2) + ∂ψ2 φ(1 - φ)(2χacψ2 - 2χccψ1) ∂2f (ψ1,ψ2,φ) ∂ψ12 ∂2f (ψ1,ψ2,φ) ∂ψ22
) φ f ′′(ψ1) + φ(1 - φ)2χca
) (1 - φ) f ′′(ψ2) + φ(1 - φ)2χac
∂2f (ψ1,ψ2,φ) ) -2χccφ(1 - φ) ∂ψ1∂ψ2 ∂2f (ψ1,ψ2,φ) ) -2χccφ(1 - φ) ∂ψ2∂ψ1
(6)
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 1e-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 self-consistent solutions to the set of eqs 2-6 in what follows:
When the crystalline mixture is completely immiscible in the solid crystalline state and forms separate crystals, the crossinteraction 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 nonideal solid solution. On the other hand, if the system is a completely ideal solid solution, the crystalline mixture would be completely miscible at all concentrations, i.e., cω ) 1, and then the crystal order parameters must be coupled to each other so that the systems can undergo cocrystallization. In the case of an intermediate situation when the crystal-crystal 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 with the heat of crystallizations of a typical polymer pair ∆H1c ) 12 kJ/mol and ∆H2c ) 16 kJ/mol, having the equilibrium melting temperatures of Tm,10 ) 500 K and Tm,20 ) 480 K, respectively. The polymer mixture is symmetric with equal chain segments, say r1 ) 10 and r2 ) 10, and the nonideality (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 2, the crystal order parameters representing the crystal-melt transition drop off discretely at a certain volume fraction as is typical for the first order phase transition. This phase transition line 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 4c. Next we determine the coexistence curves by balancing the chemical potentials of the constituents in each phase. Figure 3 illustrates the variation of the free energies of the eutectic crystallization in which the solid crystal-amorphous liquid and the amorphous liquid-solid crystal curves are intersecting with the single free energy well of the isotropic liquid (470 to 450 K). The points of intersection 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 coexistence of two crystal solid phases (Figure 3). At 440 K, the two crystal-amorphous phase transitions emerge to a eutectic point exhibiting the dual minimum free energy wells to suggest that the two crystal solids are in equilibrium with the isotropic liquid (Cr-L-Cr). These free energy curves were utilized in the establishment of the eutectictype phase diagrams. If the system were undergoing liquidliquid-phase separation, such a nonideal liquid solution may be represented by a double-well. Figure 4a illustrates the self-consistent solution with cω ) 1 showing an ideal solid solution phase diagram with a crystalliquid coexistence region bound by the liquidus and solidus lines.
16062 J. Phys. Chem. B, Vol. 110, No. 32, 2006
Figure 2. The variation of the crystal order parameters of the constituents with volume fraction at various temperatures. The parameters utilized where r1 ) 10, r2 ) 10, cω ) 0.65, ∆H1c ) 12 kJ/mol, ∆H2c ) 16 kJ/mol, Tm,10 ) 500 K, and Tm,20 ) 480 K.
Figure 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 value of cω < 1 implies that the crystallization is favored to occur as separate individual crystals and vice versa, while cω > 1 indicates the formation of azeotropic cocrystals which will be discussed later. Panels b and c of Figure 4 illustrate the development of various topologies of phase diagrams with decreasing cω in which the crystal-crystal coexistence curve is pushed upward while the crystal-liquid coexistence region is thrust downward (Figure 4b). At the same value of cω ) 0.65, the two curves intersect and form a tricritical point which is also known as the eutectic point in the phase diagram (Figure 4c), displaying the crystal-liquid and crystal-crystal coexistence regions. It should be emphasized that the present calculation was undertaken under the assumption that the isotropic
Matkar and Kyu liquid 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 Figure 4d, the binodal curve for liquidliquid 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 crystal phase and two liquid phases. Such crystalline systems are of interest not only from the thermodynamic point of view, but also from the growth dynamics of the microstructures that involve interplay between solidification dynamics of crystalcrystal, 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 (CrCr-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 5a). 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 crystalliquid, 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 melting 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 nonideal solid solution as well as the case of nonideal liquid and nonideal solid solution. The azeotrope is defined as a blend where the crystal solid solution (cocrystal) is more 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 6a shows the development of the azeotrope as a consequence of increasing the miscibility between the twocrystal 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 Figure 6a. As depicted in Figure 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 where the crystal phase is in equilibrium with two liquid phases, i.e., the L-Cr-L line. Comparison with Experiment To justify some of the predictions afforded by the present theory, it is instructive to compare the self-consistent solution with the experimental phase diagrams of the binary crystalline polymer blends, if available. We are fortunate to notice the
Binary Crystalline-Crystalline Polymer Blends
J. Phys. Chem. B, Vol. 110, No. 32, 2006 16063
Figure 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) nonideal solid solution, cω ) 0.85; and (c) eutectic phase diagram at cω ) 0.65. (d) When the nonideality 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.
Figure 5. (a) the phase diagram of an ideal liquid solution, but nonideal solid solution with χaa g χcrit leads to the formation of a peritectic (Cr-Cr-L) and (b) the nonideal liquid solution and nonideal solid solution with χaa . χcrit result in the formation of monotectic (L-L-Cr) above the peritectic line.
intriguing phase diagrams reported by Smith and Manley5 for two polyethylene fractions of different molecular weight, denoted 1000 (PE1000) and 2000 (PE2000), respectively. Thermal characterization and X-ray diffraction revealed the existence of a common crystalline lattice in which both constituents crystallize simultaneously. Figure 7a 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 experimental conditions reported by Smith and
16064 J. Phys. Chem. B, Vol. 110, No. 32, 2006
Matkar and Kyu
Figure 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 components. (b) When the liquid-liquid mixing is nonideal, i.e., χaa . χcrit in the vicinity of the transition temperatures, there is a formation of a L-L coexistence curve above the Cr-L coexistence curves.
Figure 7. (a) Solid solution of polyethylene fractions in comparison with the experimental data of Smith and Manley;5 (b) PCL/trioxane eutectic phase diagram6 showing the presence of crystal-liquid and crystal-crystal regions bound by the liquidus and solidus lines.
Manley,5 viz., ∆HPE2000c ) 3.68 kJ/mol, ∆HPE1000c ) 3.80 kJ/ mol, Tm,PE20000 ) 378.2 K, Tm,PE10000 ) 393.7 K, r1 ) 70, and r2 ) 128. The observed single melting transition indicates that the crystal solid solution must be ideal, and thus the crossinteraction term is taken as unity, i.e., ω12 ) 1. The two lines exhibiting convex and concave curvatures form the liquidus and solidus loop (Figure 7a). The experimental melting transition points are closer to the liquidus line. Although the solidus curve is identifiable in the self-consistent simulation, it is difficult to detect 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 Manley6 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 334 K, whereas trioxane is a high melting solvent that has a melting temperature of 335 K. In Figure 7b is shown the eutectic phase diagram of the PCL/trioxane mixture solved self-consistently by using the material and experimental parameters reported by Wittman and Manley,6 viz., ∆HPCLc ) 4.14 kJ/mol, ∆Htrioxanec ) 3.4 kJ/mol, Tm,PCL0 ) 334 K, Tm,trioxane0 ) 335 K, r1 ) 1400, and r2 ) 1. It is assumed that the liquid solution is completely ideal, whereas the crystal solid solution is highly nonideal, i.e., ω12 ) 0.1.
Wittman and Manley6 solved these melting point depression curves in the context of the Flory diluent theory by treating one constituent at a time without the interference by 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.6 Again the calculated solidus lines are located very close to or coincided 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 self-consistent solutions with the reported phase diagrams because the equilibrium phase diagrams are rare to come by 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. The present paper is the first to demonstrate the important roles of the crystal-amorphous, amorphouscrystal, crystal-crystal interactions in the establishment of the eutectic, peritectic, and azeotropic phase diagrams of binary
Binary Crystalline-Crystalline Polymer Blends
J. Phys. Chem. B, Vol. 110, No. 32, 2006 16065
crystalline polymer blends which are consistent with other materials phase diagrams such as metal alloys and liquid crystal mixtures.
Acknowledgment. The present study is made possible by the support of the National Science Foundation through the Grant Nos. DMR-0209272 and DMR-0514942.
Conclusions
References and Notes
Various phase diagrams including eutectic, peritectic, and azeotropic phase diagrams of a two-crystalline polymer blend have been established in the framework of a unified theory by taking into consideration all possible interactions such as amorphous-amorphous, crystal-amorphous, amorphouscrystal, and crystal-crystal interactions, provided that there is no specific interaction such as ionic or ion-dipole interaction or hydrogen bonding. The self-consistent 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 phase diagrams of polyethylene fractions as well as that of PCL/ trioxane mixtures of Manley and co-workers.5,6 It should be emphasized that the present paper is the first to incorporate the anisotropic interactions such as crystal-amorphous, amorphouscrystal, and crystal-crystal interactions in the theoretical descriptions of the equilibrium phase diagrams of binary crystalline polymer blends. It is also the first to point out the need for the solidus line in the crystalline polymer phase diagrams, which has been neglected or not captured in the polymer literature.
(1) Wheeler, A. A.; Boettinger, W. J.; McFadden, G. B. Phys. ReV. A 1992, 45, 7424-7439. (2) Kerridge, R. J. Chem. Soc. 1952, 4577-4579. (3) Koningsveld, R.; Stockmayer, W. H.; Nies, W. Polymer phase diagrams; Oxford University Press: Oxford, UK, 2001. (4) Smith, P.; Pennings, J. J. Polym. Sci Polym. Phys. Ed. 1977, 15, 523-540. (5) Smith, P.; Manley, R. S. J. Macromolecules 1979, 12, 483-491. (6) Wittmann, J. C.; Manley, R. St. J. J. Polym. Sci., Polym. Phys. Ed. 1977, 15, 1089-1100. (7) Flory, P. J. J. Chem. Phys. 1947, 15, 684. (8) Drolet, F.; Elder, K. R.; Grant, M.; Kosterlitz, J. M. Phys. ReV. E 2000, 61, 6705-6720. (9) Gunton, J. D.; Miguel, M. S.; Sahni, P. S. In Phase Transitions Crit. Phenom. 1983, 8, 267-466, 479-82. (10) Harrowell, P. R.; Oxtoby, D. W. J. Chem. Phys. 1987, 86, 29322942. (11) Chan, S. K. J. Chem. Phys. 1977, 67(12), 5755-5762. (12) Kyu, T.; Mehta, R.; Chiu, H. Phys. ReV. E 2000, 61, 4161-4170. (13) Xu, H.; Matkar, R.; Kyu, T. Phys. ReV. E 2005, 72, 011804. (14) Flory, P. J. J. Chem. Phys. 1941, 9, 660-1. (15) Flory, P. J. J. Chem. Phys. 1942, 10, 51-61. (16) Huggins, M. L. Int. ReV. Sci.: Phys. Chem., Ser. Two 1975, 8, 123152. (17) Chiu, H.; Kyu, T. J. Chem. Phys. 1998, 108, 3249-3255. (18) Chiu, H.; Kyu, T. J. Chem. Phys. 1995, 103, 7471-7479.
