Polymer 45 (2004) 6415–6424 www.elsevier.com/locate/polymer A method for determination of thermodynamic and solubility parameters of polymers from temperature and molecular weight dependence of intrinsic viscosity A.E. Bozdogan* Department of Chemistry, Yıldız Technical University, 34210 Esenler, Istanbul, Turkey. Received 28 January 2004; received in revised form 20 April 2004; accepted 22 April 2004 Abstract An equation using the temperature dependence of intrinsic viscosity of a polymer was proposed for the determination of the partial molar entropy and enthalpy changes of the polymer for mixing in dilute solution. It was found that the partial molar entropy change of a polymer for mixing at a given temperature is proportional to the hydrodynamic volume or segment number of the polymer. The partial molar enthalpy change of the polymer for mixing was determined from the thermodynamic equilibrium property of polymer phases. The solubility or cohesion parameter of a polymer fraction was calculated by using the partial molar enthalpy change and repeat unit volume of the polymer. The solubility parameter of high molar mass polymer at a given temperature was determined by extrapolating solubility parameter values of polymer fractions to high molar mass by using the solubility parameter-segment number relation of polymer fraction. This relation gives a straight line. The solubility parameter of the polymer at a given temperature and the effective interchange energy parameter for polymer– solvent pair were obtained directly from the intercept and the slope of this line, respectively. These equations were applied to the intrinsic viscosity–temperature data of polystyrene fractions in decalin solutions, and polystyrene fractions in decalin, cyclohexane and dioctyl phthalate solutions at the theta temperatures and in toluene solutions at the given temperatures. The results obtained in this study coincide with the literature values. In addition, it was given a relation, which is derived from the blob theory for the temperatures above the theta point, for the estimation of the thermodynamic parameters of polymers for mixing. q 2004 Elsevier Ltd. All rights reserved. Keywords: Intrinsic viscosity–temperature relation and thermodynamic parameters; Solubility parameter–segment number relation; The blob theory and thermodynamic parameters 1. Introduction a polymer for mixing in dilute solution. Also, the equations using the hydrodynamic volume or segment number of a The entropy and enthalpy changes of a polymer for polymer fraction for determination of the partial molar mixing determined by the thermodynamic theories [1–5] of entropy change for mixing at a given temperature will be polymer solutions are not satisfactory. Therefore, the proposed. The partial molar enthalpy change for mixing is solubility parameter of a polymer cannot be determined determined from the thermodynamic equilibrium property directly from the experiment [6] and several indirect of polymer phases. The solubility parameter of a polymer methods have been employed. fraction is calculated from the partial molar enthalpy change In our previous study [7], a method was proposed for the and the repeat unit volume of the polymer. Subsequently, determination of thermodynamic and solubility parameters the solubility parameter of the polymer is determined by of polymers from critical volume fractions. In this study, an extrapolating solubility parameter values of the polymer equation using the temperature dependence of intrinsic fractions to high molar mass by using the solubility viscosity of a polymer will be proposed for the determi- parameter–segment number relation. A linear relationship nation of the partial molar entropy and enthalpy changes of is obtained. The solubility parameter of the polymer and the effective interchange energy parameter for the polymer– * Tel.: þ90-212-449-18-33; fax: þ90-212-224-50-13. E-mail address: [email protected] (A.E. Bozdogan). solvent pair introduced in the equation of state theory [2] are 0032-3861/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymer.2004.04.065 6416 A.E. Bozdogan / Polymer 45 (2004) 6415–6424 obtained directly from the intercept and the slope of this obtained: line, respectively. ½h 2 ½h DH DS In addition, the Han Eq. [8] which is derived from the ln L ¼ 2 2 2 ð9Þ blob theory for the asymptotic behavior of intrinsic viscosity ½h RT R at the good solvent limit is modified and used for the ; where DG2 DH2 and DS2 are the partial molar Gibbs free estimation of the thermodynamic parameters of polymers energy, enthalpy and entropy changes for mixing of a for mixing. polymer fraction, respectively. A plot of ln{ð½hL 2 ½hÞ=½h} versus 1=T will give a straight line with a slope = = : of DH2 R and an intercept of 2DS2 R 2. Theory Then, the cohesion or solubility parameter d of a polymer fraction can be calculated from the following equation. Mixing solvent (S) and polymer (P) can be represented = u 1=2 schematically as a chemical equilibrium reaction with d ¼ðDH2 V Þ ð10Þ intermolecular interactions shown as dotted line [5]: where Vu is the volume per molar structural or repeat unit. The relationship between ½h and M is given by the S þ P ¼ S…P ð1Þ Mark–Houwink equation. ½ ¼ a ð Þ The equilibrium constant for this reaction can be written h KM 11 as, where K and a are constants for a given polymer–solvent– temperature system. It was shown that a ¼ 0:5 under theta 0 ½S…P K ¼ ð2Þ conditions, and that a increases to a limiting value of 0.8 ½S½P with coil expansion [9]. If Eq. (11) is multiplied by M; the where ½S is the solvent, ½S…P and ½P are the mixed and following equation is obtained. unmixed polymer concentrations in molar units. In dilute ½hM ¼ KMaþ1 ð12Þ polymer solutions, however, the concentration of solvent is large compared with the concentration of polymer and can where ½hM is the hydrodynamic volume of the polymer [9]. be considered to be constant. Therefore, Eq. (2) can be If x is taken as the number of segments of a polymer, Eq. written as, (12) can be written as 1þa 1þa 0 ½S…P ½ ¼ ð Þ K ¼ K ½S¼ ð3Þ h M KMs x 13 ½P where Ms is the molecular weight of a segment. At the theta In the mixing process, the measured intrinsic viscosity condition, from Eq. (13) the following equations ½h is proportional to the mixed polymer concentration. ½h M u ¼ 3=2 ð Þ 3=2 xu 14 ½h¼f ½S…Pð4Þ KuMsu Then, for the unmixed polymer concentration the and following equation can be written. 3=2 M 3=2 ¼ xu ð15Þ ½hL 2 ½h¼f ½Pð5Þ Msu where ½hL is the limiting intrinsic viscosity, which can be written, where Msu is the molecular weight of a corresponds to the maximum swelling of the polymer and segment and xu is the number of segments at the theta f is the proportionality constant. temperature, respectively. Substituting Eqs. (4) and (5) into Eq. (3) gives It was shown in our previous study [7] that the partial molar entropy change of a polymer for mixing at the theta ½h K ¼ ð6Þ condition is given by the following equation. ½hL 2 ½h = DS2u ¼ 2R lnðf2u xuÞð16Þ According to the second law of thermodynamics, the following equations Applying the thermodynamic criteria for the theta composition to the Flory–Huggins model yields [5]. DG ¼ 2RT ln K ð7Þ 2 = 1=2 f2u ¼ 1 ð1 þ xu Þð17Þ and For high molecular-weight polymer xu @ 1; the Eq. (17) DG2 ¼ DH2 2 TDS2 ð8Þ can be written as = 1=2 are written. From Eqs. (6)–(8), the following equation is f2u ¼ 1 xu ð18Þ A.E. Bozdogan / Polymer 45 (2004) 6415–6424 6417 ; : Substituting Eq. (18) into Eq. (16), the following temperature T ¼ Tc x can be substituted xc According to = 1=2 = = : equations the Flory–Huggins model [5], xc ¼ 1 xc þ 1 2xc þ 1 2 Then, 1=T should be linear with ð1=X Þð1=x1=2 þ 1=2x Þ: It ¼ 3 ð Þ c 12 c c DS2u 2R ln f2u 19 was shown in the previous study [7] that d2 is proportional ; = 2 = and to Tc therefore, 1 dc should be a linear function of ð1 X12Þ = 1=2 = : = 2 = ð1 xc þ 1 2xcÞ At the theta temperature, 1 dc and xc can be DS ¼ R lnðx3 2Þð20Þ = 2 : 2u u substituted by 1 du and xu Thus, the du values of polymer are obtained. Substituting Eqs. (14) and (15) into Eq. (20) fractions at the theta temperature can be extrapolated to high ; = 1=2 = ; gives molar mass of polymer du1 in which 1 xu @ 1 2xu by ! means of the following equation. ½h M ¼ u ð Þ = 2 = 2 = ^ = 1=2 DS2u R ln 3=2 21 1 du ¼ 1 du1 þð1 X 12Þð1 xu Þð24Þ KuMsu where the parameter X^ is proportional to the effective and 12 interchange energy parameter X 12 and can be taken as the = 3=2 = 2 = 1 2 M estimate of that parameter. A plot of 1 du versus 1 xu will DS2u ¼ R ln ð22Þ = ^ M give a straight line with a slope of 1 X12 and an intercept of su = 2 : 1 du1 equations. In a good solvent, the partial molar entropy change DS2g The temperature composition phase diagrams for poly- for mixing of a polymer fraction can be obtained from mer solutions show that two phases are at equilibrium at the ½hM theta u temperature. Then, the following equation can be ¼ ð Þ DS2g ln 1þa 25 written KMs and DH2u ¼ uDS2u ð23Þ M 1þa where DH2u is the partial molar enthalpy change of a DS2g ¼ Rln ð26Þ polymer for mixing at the theta temperature.