Relationship between the Glass Transition Temperature and the Interaction Parameter of Miscible Binary Polymer Blends

An equation that relates the g h transition temperature, TE, and a binary interaction parameter, x, for miscible binary polymer blends was derived. The equation including no adjustable parameters was based on a thermodynamic mixing formalism using enthalpy as the thermodynamic parameter. The enthalpy of mixing was written as a van Laar expression, and the T, was formally treated as a second-order Ehrenfest transition. The equation satisfactorily predicts TK-composition curves for miscible bw polymer blends that exhibit either positive or negative deviation from a linear mixing rule, depending on the strength of the interaction. Good agreement was found between calculated x and values experimentally determined by equilibrium melting point depression and inverse gas chromatography. Introduction Specific interactions between polymers have a profound influence on the phase behavior of binary polymer blends. To form a miscible blend, the free energy of mixing A G m I 0 and the second derivative with respect to composition d2AGJad2 > 0, where (1) and A H m and AS, are the enthalpy and entropy of mixing, respectively. For high molecular weight polymers, A S m is negligibly small and the sign of AG, is dominated by A H m . In general, AH, is negative only if there are specific associative interactions between the two polymers. Therefore, the formation of miscible polymer blends depends on the occurrence of exothermic interactions such as hydrogen bonding,' dipole-dipole interactions,2 acid-base interactions,3 or transition metal comple~ation.~ A common method used to judge the miscibility of blende is to measure the glass transition temperature(s), TP Two Tg(s indicab a two-phase syatem, and a single compositiondependent Tg is often taken as evidence of the formation of a miscible blend. In practice, one would like to be able to predict the Tg of a miscible blend from the properties of the component polymers. Experimental studies on the composition dependence of the glass transition temperature for a variety of binary miscible blends show both negative and positive deviations from a simple weighted average (i.e., a rule of mixtures) of the two Tis of the component polymers. Over the past 40 years, a number of theoretical equations have been proposed to predict the composition dependence of T, for miscible blends. The most frequently cited equations are those proposed by Gordon and Taylor: Fox? Couchman? and Brekner et al.8 With the exception of the last treatment, these equations are only able to predict Tgcomposition curves that exhibit negative deviation from the rule of mixtures prediction. Positive deviation occurs in systems with strong intermolecular interactions, and the failure of the predictions is due to the inability of these equations to account for strong interaction^.^ The equation of Brekner et al.8 can predict positive deviation, but it contains two adjustable parameters that cannot be independently determined. Prud'homme'O and Lin et al." correlated qualitatively the fitting parameters K in the Gordon-Taylor equation (5) or q in Kwei's equation12 with the strength of the interactions in binary blends. Since K and q are adjustable AG, = AHm TAS, 0024-9297/92/2225-3242$03.00/0 empirical parameters, however, the ability to extract fundamental information by this approach is limited. More recently, Couchman13 derived a Tg equation which considers the effect of the strength of interactions using the quasi-chemical approximation. Like the equation of Brekner et al.? however, Couchman's interaction term contains two adjustable parameters. Building upon Couchman's theory for the composition dependence of Tg,14 Painter et al.15 recently incorporated the effect of hydrogen bonding on the enthalpy of mixing and the composition dependence of TP Knowledge of the equilibrium constants of self-association and association between unlike groups in a blend exhibiting hydrogen bonding allows one to calculate the enthalpy of mixing and TP It is desirable to have a scheme for correlating Tg with a measure of the intermolecular interactions, such as the Flory-Huggins interaction parameter, x . In this paper, we extend Couchman's theory14 to include a term involving x . When x is known from another independent measurement, such as from melting point depression data, this theory can be used to predict TP Alternatively, when the Tg-composition relationship is known, the theory can be used to calculate x . Theoretical Considerations The glass transition can be treated formally as an Ehrenfest transition of second order.16 The thermodynamic functions, entropy, enthalpy, and volume, are continuous at Tg, but the first derivatives of these functions undergo discontinuities at Tg. In principle, any of these thermodynamic functions could be used to derive the relation for the compositional variation of Tg, but, in practice, the dependence of these variables on composition is not usually known. As a result, approximations of the thermodynamic functions are usually ~ 5 e d . l ~ Owing to the importance of enthalpic interactions on the phase behavior or polymer blends and the relationship between enthalpy and x , the theory described herein employed enthalpy as the thermodynamic variable. Consider two polymers having glass transition temperatures Tgl and Ta and respective molar enthalpies HI and Hz. The molar enthalpy of a mixture of the two polymers is (2) where xi represents the mole fraction of polymer i in the blend and AHm is the excess enthalpy of mixing. The Hmix = x,H, + xa2 + AH,