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Flory Enthalpy Parameter at Infinite Dilution of Polymer Solutions Determined by Various Methods

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Flory Enthalpy Parameter at Infinite Dilution of Polymer Solutions Determined by Various Methods Polymer Journal, Vol. 20, No. I, pp 31--43 (1988) Flory Enthalpy Parameter at Infinite Dilution of Polymer Solutions Determined by Various Methods Kenji KAMIDE, Shigenobu MATSUDA, and Masatoshi SAITO Fundamental Research Laboratory of Fibers and Fiber-Forming Polymers, Asahi Chemical Industry Co., Ltd., Hacchonawate 11-7, Takatsuki, Osaka 569, Japan (Received July 16, 1987) ABSTRACT: The Flory enthalpy parameter at infinite dilution Ko was evaluated for atactic polystyrene (PS)--cyclohexane (CH) and -trans-decalin (D) systems by applying the following four methods from literature data: (I) Temperature dependence of vapor pressure and membrane osmotic pressure, (2) critical solution temperature Tc and critical solution concentration v; for a series of solutions of polymers, (3) temperature dependence of the second virial coefficient in the vicinity of the Flory theta temperature 0, and (4) calorimetry. In method 2, Shultz-Flory, Stockmayer, Koningsveld et al. (KKS), and Kamide--Matsuda (KM)'s procedures were applied to estimate 0 and the entropy parameter t/J0 , from which Ko was calculated as t/100/T (T, temperature). Ko values at 0 deduced by these methods (in method (2), by KKS and KM treatments) yielded a single master curve against the weight- or number-average molecular weight Mw or M.; Ko (at 0)=0.924.Mw -o.os9 for PS---CH system. Excellent agreement (±0.02) was confirmed between Ko values at 0 deduced by various methods when M w ( or M.) of the polymer was the same. This fact strongly supports the validity of the modified Flory-Huggins solution theory, in which the polymer-solvent interaction parameter x depends on the polymer concentration and its molecular weight. KEY WORDS Flory Enthalpy Parameter/ Polymer Solution/ Polystyrene/ Chemical Potential / Osmotic Pressure / Light Scattering / Second Virial Coefficient / Critical Solution Point / Calorimetry / The pair interaction Flory enthalpy param­ K=Ko+K1Vp+K2v/+ ... (2) eter K is defined by the van Laar-Scatchard type relation1 : Ko in eq 2 is given by (1) K 0 = lim {AH0 /(RTv/)} (3) vp-+O where AH0 is the partial molar heat of dilution with respect to the solvent, R, the gas constant, Theoretically, Kor simply Ko can be evaluated T, Kelvin temperature, vp, the polymer volume by various methods, directly or indirectly as fraction. will be described later. In other words, if good Calorimetry allows direct determination of agreement between K or Ko values evaluated by AH0, and thus K. The calorimetric experi­ various methods is confirmed, the theory, on ments, made by Fujishiro and his students, which the principles of the methods are based, indicated K not to be constant independent of is considered thoroughly acceptable to explain T and vP, which was first assumed in the all the thermodvnamic properties of polymer original Flory-Huggins theory, but a purely solutions. phenomenological parameter, which depends In 1954, Krigbaum3 found that the values of on both T and v/. the entropy parameter t/10 (see eq 7) (thus, the 31 K. KAMIDE, s. MATSUDA, and M. SAITO enthalpy parameter Ko at theta temperature very small partial heats of dilution and have (see eq 6)) for atactic polystyrene (PS)­ relatively larger viscosities, cyclohexane (CH) system deduced from the (4) it was confirmed that any thermody­ critical solution points4 (by his expression, the namic parameter can not be accurately de­ precipitation temperature) by Shultz-Flory termined from hydrodynamic properties in­ method (eq 8) and from intrinsic viscosity5 cluding the limiting viscosity number for lack differ from those obtained from the osmotic of sound theoretical background. second virial coefficient A 2 by factors of about In this article we calculated Ko for atactic 3 and 1/3, respectively. He concluded that the PS-CH and trans-decalin (D) systems by magnitude of the important entropy parameter applying various direct or indirect methods to t/1 0 remains in doubt pending on further refine­ literature data available in order to clarify ment of existing theories. Thereafter (in 1955), whether Ko values thus evaluated coincide with Krigbaum and Carpenter6 attempted to settle each other. this problem by estimating t/1 0 from the tem­ perature dependence of the light scattering THEORETICAL BACKGROUND second virial coefficient but without success because they obtained for the same polymer­ Ko can be determined by the following solvent system a surprisingly low t/10 , which ,nethods: was considered mainly due to large experimen­ tal error inherent in the light scattering (I) The temperature dependence of vapor measurements. pressure and osmotic pressure through use of Up to now, as far as authors know, no the relations1 further attempt to compare t/10 or Ko values Ko= lim {l/(RTv/)}{o(il11o/T)/o(l/T)}r,Vp estimated by various methods has been made. vp--+O Note that for this purpose Ko is preferable to (4) t/10 , because the former can be determined directly without using any particular theory of and a polymer solution. ilt1o = RTln(P0 / P0 °) = - n/ V0 ° (5) We believe that for the past 30 years the situation has drammatically changed and a Here, ilt1o is the chemical potential of the great deal of progress has been made and solvent in solution, P0 , the vapor pressure of among the latest developments are the follows: the solvent component in solution, P0 °, the (1) the reliability of A2 determination by vapor pressure of the pure solvent, n, the the light scattering (LS) method was improved osmotic pressure, V0 °, the molar volume of the remarkably as a result of well-established ex­ solvent. The partial differentiation of !lµ0/T perimental techniques and instrumental ad­ with respect to 1/T is carried out under con­ vance (for example by using laser light) and a stant pressure and constant composition ex­ sufficient accumulation of the experimental cept the polymer. data with high degree of precision for mono­ disperse polymer solutions has been made very (2) The critical phenomena (critical solution systematically, temperature Tc and critical polymer concen­ (2) prominent progress was achieved in tration v/). Ko are related to Flory theta theory on the critical phenomena of polymer temperature 0 and Flory entropy parameter t/10 solutions, through the definition of 0. 1 (3) advance of calorimetry allowed us to (6) apply it to the polymer solutions, which have 32 Polymer J., Vol. 20, No. I, 1988 Flory Enthalpy Parameter of Dilute Polymer Solutions with Here Xz is the z-average X. Plot of 1/Tc as a function of {(l/Xw112 +(Xz/Xw)1i2)(1/Xw1i2 i/1 0 = lim (ilS0 - ilS0 comb)/(Rv/) 112 )) VpO-+Q +(Xw/Xz) )}/2 yields (1/(01/10 from slope and (1-1/(21/10))/0 as an intercept at = lim (Llµ 0 - LlH O - T !lS0 comb)/(RTv/) vp0 -o {(l/Xwl/2 +(Xz/Xw)l/2)(1/Xwl/2 + (7) (Xw/ Xz) 112 )}/2 = 1/2. ilS is the partial molar entropy of dilution 0 Therefore, 0 and 1/1 can be unambiguously and ilS comb, the combinatorial entropy term. 0 0 determined. 0 and l/10 can be evaluated from Tc (and vp") for a series of solutions of polymers having (c) Koningsveld et al. (KKS) Method8 different molecular weights by the following An alternative expression of neutral equilib­ four methods: rium conditions is given by (a) Shultz and Flory (SF) Method4 Y=g1 -g2+4g2v/ Shultz and Flory derived a relation between (11) Tc and the weight-average molar volume ratio of the polymer to the solvent X,.. given by with (12-a) l/Tc=(l/01/Jo)(l/Xw112 + 1/(2Xw))+ 1/0 (8) The slope of the plot of 1/Tc against (l/X/12 =(1/vo) f-vo xd(l-vo) (12-b) + l/(2Xw)) gives 1/01/10 and its intercept gives 1/0. It must be noticed that eq 8 is strictly valid only for a monodisperse polymer-single sol­ where g is a thermodynamic interaction pa­ vent system. Furthermore, the x-parameter is rameter, related to x-parameter through eq given in a well-known relation 12-b, v0 , volume fraction of the solvent ( = 1-vP). g1 and g2 in eq 11 are calculated Llµ0 = RT {ln(l -vP) (9) from vPC, Xw, and Xz by applying the curve +(1-1/Xn)vp+xv/} fitting method to eq. 11. g0 , defined in eq 12-a, is evaluated from g1 (Xn is the number-average molar volume ratio and g2 , and v/ and Xw, using the spinodal of the polymer to the solvent) and is assumed equation, to be independent of the polymer molecular weight and the polymer concentration. 2go = 2(goo + god TJ = 1/(1-vp°)+ 1/(v/Xw) (b) Stockmayer Method7 Stockmayer derived a relation (eq 10) for +2g1(1 -3vp°) +6gi(l -2v/)vp° (13) multi-component polymers-single solvent sys­ Plot of g , obtained thus, versus T - 1 enables tem, in which the x parameter is independent 0 0 us to estimate g and g01 . Considering the of vP and molar volume ratio of the polymer 00 relation to the solvent X. (14) 1/Tc =(1/(201/Jo)){(l/X,..1/2 +(Xz/X,.,)112) X ( 1/ X wl/2 + (X,../ Xz)l/2)} we obtain the following relations: 1/10 (15) +(1-1/(21/10 ))/0 (10) = 1/2-goo+Yoi 0=go1No (16) Polymer J., Vol. 20, No. 1, 1988 33 K. KAMIDE, s. MATSUDA, and M. SAITO (3) Temperature dependence of A 2 by mem­ K0 =g0 i/T (17) brane osmometry or light scattering measured Then, Ko can be determined from eq 17. in the vicinity of 0 temperature (eq 22) via 0 and 1/1 0 •2 (d) Kamide-Matsuda Method9 t/1 =(V0°/ii)0(aA2/aT)6 (22) At the critical point, the following equations 0 can be derived,9 Here v is the specific volume of polymer and note that 1/10 estimated by eq 22 corresponds to the finite molecular weight.
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