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Macromolecular Research, Vol. 16, No. 4, pp 320-328 (2008)

Consideration of Long and Middle Range Interaction on the Calculation of Activities for Binary Polymer

Seung Seok Lee, Young Chan Bae*, and Yang Kook Sun Division of Chemical Engineering and Molecular Thermodynamics Laboratory, Hanyang University, Seoul 133-791, Korea Jae Jun Kim College of Architecture, Hanyang University, Seoul 133-791, Korea

Received October 29, 2007; Revised December 26, 2007; Accepted January 2, 2008

Abstract: We established a thermodynamic framework of group contribution method based on modified double lat- tice (MDL) model. The proposed model included the long-range interaction contribution caused by the Coulomb electrostatic forces, the middle-range interaction contribution from the indirect effects of the charge interactions and the short-range interaction from modified double lattice model. The group contribution method explained the com- binatorial energy contribution responsible for the revised Flory-Huggins entropy of mixing, the van der Waals energy contribution from dispersion, the polar force, and the specific energy contribution from hydrogen bonding. We showed the activities of various polymer systems in comparison with theoretical predictions based on experimental data. The proposed model gave a very good agreement with the experimental data. Keywords: modified double lattice model, group contribution method, charge interaction, long range interaction, mid- dle range interaction.

Introduction et al.,9 Kontogeogis et al.,10 and Bogdanic and Fredenslund.11 Those methods are based on the UNIFAC correlation For engineering purposes, it is often necessary to make which is often successful for estimating phase equlibria in estimate of activity for where only fragmentary mixtures containing ordinary (nonpolymer) liquids. data, or no data at all, are available. For vapor-liquid equi- The fundamental basis for existing group-contribution libria, such estimates can be made using a group-contribu- methods for polymer solutions is the lattice theory of Flory12 tion method.1 and Huggins.13 To pursue a formal exact solution to the lat- Since activity coefficients calculation using group-contri- tice model using advanced statistical and mathematical butions was suggested by Langmuir,2 the most widely used methods, Freed and coworkers14,15 developed a lattice-clus- and best known of group-contribution method is the UNI- ter theory for polymer-solvent system. This theory provides FAC.3 The acronym UNIFAC denotes the UNIQUAC4 an exact mathematical solution for the Flory-Huggins functional group . The UNIFAC correla- model. tion is based on a semi-empirical model for liquid polymer Hu et al.16 proposed a new theory called the double lat- solutions called UNIQUAC (universal quasi-chemical activity tice model based on Freed’s lattice-cluster theory and Bae coefficient). When compared with the experimental data, et al.17 reported a modified double-lattice model and pro- however, the UNIFAC equation shows deviations too large vided an exact mathematical form for the secondary lattice to satisfactorily explain the polymer solutions. of the double lattice. Hu et al.18 presented the group contri- Oishi and Prausnitz5 modified the UNIFAC model by pro- bution method including a revised Flory-Huggins entropy viding a free contribution suggested by Prigogine- and a series expression for excess internal energy as well Flory-Patterson theory for polymer solutions to consider the as a double lattice model to account for specific interac- compressibility and change in upon isothermal mixing. tions. Later, many modifications of UNIFAC model are reported Debye and Hückel19 presented the theory of inter-ionic by Holten-Andersen and Fredenslund,6,7 Chen et al.,8 Elbro attraction in aqueous electrolyte solutions. This theory first made it possible to calculate the ion activity coefficients at *Corresponding Author. E-mail: [email protected] infinite . In order to apply the Debye-Hückel theory

320 Calculation of Activities of Polymer Solutions Using Group Contribution Method Based on Modified Double Lattice Model to higher electrolyte , Stokes and Robinson20 polymer solutions. In this study, we calculate the excess proposed the concept of ionic hydration, and derived a two Gibbs energy as the sum of three contributions: parameter equation. Later several variations of the ionic 21 22 E E E E hydration theory are reported: Glueckauf, Robinson et al. G = GLR ++GMR GSR (1) and Stokes and Robinson.23 These theories are used for cal- E culation of the activity of water in concentrated aqueous GLR represents the long-range (LR) interaction contribu- electrolyte solutions. tion caused by the Coulomb electrostatic forces, and mainly 28 E In the early 1970s, the integral techniques based on the describes the direct effects of charge interactions. GMR Ornstein-Zernicke equation have been used to solve the represents the indirect effects of charge interactions. In primitive model of electrolyte solutions. Blum24 used the order to distinguish the indirect effects from the non-charge method proposed by Baxter25 for hard-sphere solutions and short-range interactions of the non-electrolyte solution, and square-well potentials, and obtained a solution for the mean because some of the charge interactions (such as the charge- spherical approximation (MSA). Planche and Renon26 gen- dipole interactions and charge-induced dipole interactions) eralized the model of Baxter by using the formalism of are proportional to r-2 and r-4, we shall refer to this term as Blum, taking into account short-range forces between - the middle-range (MR) interaction contribution. (r is a dis- 32,34 E cules in order to get a non-primitive representation of elec- tance with two particles). GSR expresses the contribution trolyte solutions. Pitzer27,28 developed general equations for of the non-charge interactions, which is identical to the the thermodynamic properties of aqueous electrolytes based short-range (SR) interactions in non-electrolyte solutions. on the as Debye-Hückel theory. However, if the Pitzer This model has two ways of usage according to the classi- parameters are applied to the high range (for fication of non-electrolyte and electrolyte: example 10-20 M) very large deviations are obtained in 1. In case of non-electrolyte, only short range interaction most cases. is considered. In last decades, numerous authors have modified the 2. In case of electrolyte, short, mid, and long range are existing gE model and EOS (Equation of State) model for considered. non-electrolyte systems. Mock et al.29 applied the NRTL Long-Range Interaction Contribution. The long-range model to electrolyte systems. Their model provides a con- electrostatic interaction explained interaction between ion- sistent thermodynamic framework for the representation of ion. It gives rise to the Debye-Hückel limiting law, which the phase equilibrium of mixed solvent electrolyte systems. states that the logarithm to the ionic activity coefficient at However, different parameters are required to calculate the high dilution is proportional to the square-root of the ionic VLE at different temperatures. Sander et al.30 proposed an strength. In all the models presented in the literature, this extended UNIQUAC model for mixed-solvent/electrolyte contribution has been described by some or other form of systems. Macedo et al.31 used a modified Debye-Hückel the Debye-Hückel theory. E term derived from the McMillan-Mayer solution theory to In this study, GLR is calculated in terms of the Debye- replace the Debye-Hückel term in the Sander model. Hückel theory as modified by Fowler and Guggenheim34: E The most widely used and best known of the g model for ion 32,33 E –1 2 2 the electrolyte solution is LIQUAC. It consists of a Debye- GLR = –()3D ∑ sizi e τ()ka (2) Hückel term to account for long-range electrostatic interac- i = 1 tions, the UNIQUAC equation for the description of short- where si is the number of i ions in the system, D is the sol- range interactions among all particles, and a middle-range vent dielectric constant, a is the distance of closest approach contribution to include all indirect effects of the charge between two ions and τ (x) is defined as interactions. 2 –3 x In this study, we propose a group-contribution model that τ()x = 3x ln()1 + x – x + ---- (3) 2 can be used to describe solvent activities of polymer solu- tions and of perfluorinated SPE/water system. where x = κa and κ is the inverse of the shielding length The proposed model is based on a modified double lattice commonly called Debye length which is a characteristic dis- model, long range interaction and middle range interaction tance of interaction. wherein the Helmholtz function of mixing includes the The expression for the long-range interaction contribution revised Flory-Huggins entropy contribution, the van der Waals to the activity coefficients of solvent, s, follows the appro- energy contribution, and the specific energy contribution. priate derivations of eq. (2). We assume the partial of the solvent s in the solution is approximated as Model Development the molar volume of the pure solvent.

Various types of interactions are taken into account for γ LR 2AMs 12⁄ 1 ()12⁄ ln s = ------3 1 + bI –21------12⁄ – ln + bI (4) describing the activities of electrolyte and non-electrolyte b 1 + bI

Macromol. Res., Vol. 16, No. 4, 2008 321 S. S. Lee et al.

where Ms is the molecular weight of the solvent s, I is an anions. and A and b are Debye-Hückel parameters. By differentiating eq. (12) with respect to the number of moles of solvent and ions, one obtains 2 I = 0.5∑mizi (5) i MR lnγ s = ∑Bsion, ()I xion – ()Ms ⁄ Mm 0.5 ion 5 d × ′ A = 1.327757 10 ------1.5- (6) []() () ′ ()DT ∑∑ Bsol, ion I + IBsol, ion I xsolmion sol ion 0.5 d − []() ′ () b = 6.359696------0.5- (7) Ms ∑∑ Bca I + IBca I mcma (13) ()DT ' ' Where xsol is the salt-free of solvent sol, B(I) is where T is the absolute temperature in Kelvin and D is the ' equal to B(I)=dB(I)/dI and Mm is the mean molecular dielectric constant for a solvent. mi is the molality of species weight of mixed solvent (kg/mol)m and is calculated as i and d is the solvent molar density. These parameters are based on a value of 4 Å for the distance of closet approach. ′ E Mm = ∑xsolMsol (14) Middle-Range Interaction Contribution. The GMR term is the contribution of the indirect effects of charge interac- Short-Range Interaction Contribution. The mostly tions to the excess Gibbs energy. For a solution containing existing model to describe the phase equilibria of non-elec- nn moles of solvent n (n=1, 2, …, sol) and nj moles of ion j 3 E trolyte solution is UNIQUAC model. However, it gives a ( j=1, 2,…, ion), GMR is obtained from the following equa- good representation only for such as hydrocarbons, ketones, tion: esters and water, etc. In this study, we employ the modified E 18 GMR double lattice model including group contribution method ------= ∑∑Bklxkxl (8) RT k l to take into account the van der Waals energy contribution, polar force and the specific energy contribution. where Bkl is the interaction coefficient between species k MDL Model. and l (ion or molecule) and xk and xl are mole fractions of 18 Primary Lattice: Oh et al. proposed a new Helmholtz species k and l, respectively. energy of mixing as the form of Flory-Huggins theory. The By using the simplest potential model for electrolyte solu- expression is given by tions and from radial distribution theories of statistical ther- Δ φ φ modynamics, one can obtain an expression for the dependence A ⎛⎞1 φ ⎛⎞2 φ χ φ φ ------= ⎝⎠----- ln 1 ++⎝⎠----- ln 2 OB 1 2 (15) of the indirect effect on the ionic strength. It seems reason- NrkT r1 r2 able to assume that Bkl, which represents all indirect effects where Nr is the total number of lattice sites and k is the Bolt- caused by the charges, can be described by the following zmann’s constant. ri is the number of segments per molecule simple equation: i. χOB is a new interaction parameter and function of ri, ε˜ : 12⁄ Bij()I = bij + cijexp()a1I + a2I (9) 2 χ ⎛⎞1 1 ⎛⎞1 ε ⎛⎞1 1 ε εφ ε2φ2 OB = Cβ⎝⎠---- – ---- ++⎝⎠2 + ---- ˜ – ⎝⎠---- – ---- + Cγ ˜ ˜ 2 Cγ ˜ 2 r2 r1 r2 r2 r1 where bij and cij are middle-range interaction parameters (16) between species i and j (bij = bji, cij = cji). a1 and a2 are con- ε˜ is a reduced interaction parameter given by stants which are determined empirically using a number of experimental data for electrolyte/solvent systems.32 These ε˜ ==ε ⁄ kT ()ε11 + ε22 – 2ε12 /kT (17) are estimated by fitting experimental polymer sorption VLE (Vapor-Liquid Equilibria) data of perfluoro-sufonated poly- where ε11, ε22 and ε12 are for the corresponding nearest- 35 mer electrolyte membranes/water. neighbor segment-segment interactions. Parameters, Cβ and Cγ , are universal constants. These constants are not adjust- 12⁄ Bion, ion = bion, ion + cion, ionexp()–0.13I + I (10) able parameters and are determined by comparing with 12⁄ Madden et al.’s Monte-Carlo simulation data (r1=1 and Bion, solvent = bion, solvent + cion, solventexp()–0.13 1.2I + I (11) r2=100). The best fitting values of Cβ and Cγ are 0.1415 and 1.7985, respectively.18 We assume that there are no middle-range interactions Secondary Lattice: In Freed’s theory,14,15 the solution of between , so that eq. (8) can be simplified to the Helmholtz energy of mixing for the Ising model is given MR Gs by ------= ∑∑Bsol, ion()I xsolxion + ∑∑Bca()I xcxa (12) RT 2 2 2 sol ion c a ΔA zε˜ x1x2 zε˜ x1x2 ------= x1lnx1 ++x2lnx2 ------– ------+ ..... (18) where c indices cover all cations and a indices cover all NrkT 2 2

322 Macromol. Res., Vol. 16, No. 4, 2008 Calculation of Activities of Polymer Solutions Using Group Contribution Method Based on Modified Double Lattice Model

SR where, z is the coordination number and xi is the mole frac- Δμ s ()φ ⎛⎞1 1 φ tion of the component i. ------= ln 1 – 2 – r1⎝⎠---- – ---- 2 kT r2 r1 To obtain an analytical expression for the secondary lat- 2 tice, we defined a new Helmholtz energy of mixing as the ⎛⎞1 1 ⎛⎞⎛⎞1 1 ε ε ⎛⎞1 ε φ2 + r1 Cβ ⎝⎠---- – ---- ++⎝⎠⎝⎠---- – ---- + Cγ ˜ ˜ ⎝⎠2 + ---- ˜ 2 fractional form to improve the mathematical approximation r2 r1 r2 r1 r2 defect by revising eq. (18). This secondary lattice is intro- − ⎛⎞⎛⎞1 1 ε ε ε φ3 εφ4 duced as a perturbation to account for the oriented interac- 2r1 ⎝⎠⎝⎠---- – ---- + Cγ ˜ ˜ + Cγ ˜ 2 + 3r1Cγ ˜ 2 (23) r2 r1 tion. The expression is given by

where φi is the segment fraction of component i, φi = Niri/Nr ΔAsec,ij 2 zCαδε˜ij()1 – η η m ------= --- ηlnη ++()1 – η ln() 1 – η ------and Nr =∑Niri is the total number of segments in the sys- NijkT z 1 + Cαδε˜ij()1 – η η i tem and ri is the segment number of component s(salt). (19) The final equation of proposed model is presented by sum of eqs. (4), (13), and (23) and is written as where, ΔAsec,ij is the Helmholtz energy of mixing of the sec- ondary lattice for i-j segment-segment pair and Nij is the Δμ1 2AMs 12⁄ 1 12⁄ number of i-j pairs, δε˜ is the reduced energy parameter ------= ------1 + bI –21------– ln()+ bI + lnxs kT 3 12⁄ contributed by the oriented interactions and η is the surface b 1 + bI ′ fraction permitting oriented interactions. For simplicity, η is +∑Bsion, ()I xion – ()Ms ⁄ Mm ∑∑[]Bsol, ion()I + IBsol, ion()I 16 ion sol ion arbitrarily set to 0.3 as Hu et al., suggested. Cα also is not ′ ′ an adjustable parameter and is determined by comparing xsolmion – Ms∑∑[]Bca()I + IBca()I mcma + lnxs with Panagiotopolous et al.’s36 Gibbs-Ensemble Monte-Carlo ()φ ⎛⎞1 1 φ simulation data of Ising lattice. The best fitting value of Cα + ln 1 – 2 – r1⎝⎠---- – ---- 2 18 r2 r1 is 0.4880. 2 Incorporation of Secondary Lattice Into Primary Lat- ⎛⎞1 1 ⎛⎞⎛⎞1 1 ε ε ⎛⎞1 ε φ2 + r1 Cβ ⎝⎠---- – ---- ++⎝⎠⎝⎠---- – ---- + Cγ ˜ ˜ ⎝⎠2 + ---- ˜ 2 tice: To incorporate a secondary lattice, we replace εij by r2 r1 r2 r1 r2 * εij −ΔAsec,ij/Nij in eq. (17). If oriented interaction occurs in − ε ε* − ⎛⎞⎛⎞1 1 ε ε ε φ3 εφ4 the i j segment – segment pairs, we replace ˜ by /kT 2r1 ⎝⎠⎝⎠---- – ---- + Cγ ˜ ˜ + Cγ ˜ 2 + 3r1Cγ ˜ 2 (24) r2 r1 +2ΔAsec,ij/NijkT in eq. (18). If oriented interaction occurs in the * i−i segment - segment pairs, we replace ε˜ by ε /kT − ΔAsec,ii/ NiikT. In this study, we assume the oriented interaction Group Contribution. occurs in the i−i, j−j and i−j segment-segment pairs. We Van der Waals Energy Contribution: The energy parame- * replace ε˜ by ter εij in eq. (17) is due to van der Waals forces (dispersion ε * Δ Δ Δ and polar forces). For a pure component i, ii can be esti- ε ()ε* ε* ε* ⎛⎞Asec,11 Asec,22 2 Asec,12 ˜ = 11 + 22–2 12 + ⎝⎠– ------– ------+ ------mated using the square of the pure-component van der N11 N22 N12 37 (20) Waals parameter of Hansen (Barton), which is

* * * the sum of a dispersion contribution and a polar contribu- where ε11 , ε22 and ε12 are van der Waals energy interaction 2 2 2 tion: δ vdw = δ d +δ p . parameters. ΔAsec,11 , ΔAsec,22 and ΔAsec,12 are the additional Helmholtz functions for the corresponding secondary lat- * 2 3NAεiiri tice. If the oriented interaction in i−i, j−j segment-segment δ vdw, i = ------(25) Vmi pairs is very smaller than interaction in i−j segment-seg- 2 ment pair, we can neglect interaction in i−i, j−j segment- where NA is the Avogadro number and where δ vdw and Vmi segment pairs. And then eq. (20) becomes are at 25 oC. For apure component, the effect of temperature * on εii is given by Δ ε˜ ()ε* ⎛⎞2 Asec,12 = –2 12 + ⎝⎠------(21) + N12 ε * ii εii = ------(26) To correlate MDL model to melting point depression the- Vmi ory, we require chemical potentials of components 1 and 2. where Vmi depends on temperature. The temperature -inde- The definition of chemical potential is + pendent parameter εii can be estimated by 2 2 o Δμi ∂Δ()A/kT + δvdwVmi()25 C ------= ------(22) εii = ------(27) kT ∂ Ni 3NAri * The final expression for the chemical potential can be The cross interaction van der Waals energy parameter εij written as is estimated by the geometric mean of the corresponding

Macromol. Res., Vol. 16, No. 4, 2008 323 S. S. Lee et al. pure-component parameters

* * * εij = εiiεjj (28)

Specific Energy Contribution: The pure-component parameters δε˜ii and δε˜jj are calculated from Hansen’s hydrogen bonding solubility parameter δh. It is related to the additional specific energy ΔUsec,ii by

o 2 o ΔUsec,ii()25 C = –δ hVmi()25 C (29)

For the temperature dependence of ΔUsec,ii, we assume + ΔUsec,ii ΔUsec,ii = ------(30) Vmi + Mw where ΔUsec,ii is independent of temperature. From eqs. (15) Figure 1. Solvent activities for the PIB (polyisobutylene, and (16), we get =1,170)/pentane system at 298.15 K. The dark circles are experi- et al 38 2 2 o mental data by Wen .. δ hVmi()25 C ΔUsec,ii() T = ------(31) Vmi()T mation methods. To establish the group-contribution method, For a pure component i, upon inserting eq. (5) into the the most significant role is to determine the cross-pair inter- ∂ ()Δ ⁄ thermodynamic relation ------AT- = ΔU , we have action between polymer and solvent segments. ∂ ()⁄ 1 T Non-Electrolyte Polymer/Solvent Systems.

⎛⎞δεii Figure 1 shows a comparison with activities of PIB (Mw NirizCα ()1 – η η ------Δ ⎝⎠ =1,170)/pentane system at 298.15 K calculated by Hu et Usec,ii k 16 ------= ------2 (32) al. and with that of this work. The solid line is calculated k δεii 16 1 + Cα ------1()– η η by this work, the dashed line is from Hu et al. and the dot- kT ted line is from this work without specific energy. Dark cir- cles are experimental data reported by Wen et al..38 Both Cross specific energy parameter δε˜ij is calculated from pair-interaction group parameters models gives fairly good agreement with experimental results. However, Hu et al.’s Model needs 18 group parame- δε Ns Np ij ters and the proposed model requires only 6 parameters. ------= ∑ ∑ φmφngmn (33) k m = 1n = 1 When result calculated from the proposed model without specific energy is compared with experimental data, it dif- where Ns and Np are number of groups in solvents and poly- fers from experimental data. Calculated group parameters mers, respectively. φm and φn are volume fractions of group m in a solvent and that of group n in a polymer, respec- from Hu et al.’s model are listed in Table I and all parame- ters from this work are listed in Table II. tively; gmn are pair interaction parameters between group m The solid line is calculated by this work. Dark circles are in a solvent and group n in a polymer. To improve the accu- 38 racy of prediction, we assume that a functional group in a experimental data reported by Wen et al.. Selecting molar ε * polymer is different from that in a solvent. at different temperatures for pure component i, ii , ε * ε * δε δε In this study, they are estimated by fitting experimental jj , ij , ii, and jj are calculated. The pair interaction vapor-liquid equilibria data of non-electrolyte polymer solutions38 and perfluoro-sufonated polymer electrolyte mem- Table I. Group-Interaction Parameters by Hu et al.13 35 branes/water. Polymer Solvent Results and Discussion CH3 CH2 C gmn (1) 81.6 -71.8 1956.6

We proposed a new group-contribution model to describe CH3 gmn (2) - -669.5 - the vapor-liquid equilibria for a variety of polymer/solvent gmn (3) - 2439.9 -8909.2 systems. The new model employs the secondary lattice con- cept to take into account an oriented interaction. The advan- gmn (1) - 663.3 -2115.6 tage of this model follows from its simplicity. In this work, CH2 gmn (2) - -1088.5 - most parameters are calculated from pure-component prop- gmn (3) 1614.5 - -9250.0 erties, either form experimental data or from published esti-

324 Macromol. Res., Vol. 16, No. 4, 2008 Calculation of Activities of Polymer Solutions Using Group Contribution Method Based on Modified Double Lattice Model

Table II. Group-Interaction Parameters: gmn(K) Polymer Solvent CH3 CH2 CH C C6H5 O

CH3 2980.1 -2493.9 -1761.2 -5486.5 -586.0 -

CH2 1512.1 -6960.2 - -5219.4 - -

C6H5 - 769.9 -3017.3 - 194.4 2845.6 H - -5776.1 - - - 1434.7 OH - -1598.9 - - - -1190.5

Figure 2. Solvent activities for the PS (polystyrene, Mw= Figure 3. Solvent activities for the PEO (poly(ethylene oxide), 10,920)/toluene system at 321.65 K. The dark circles are experi- Mw=100,000)/benzene system at 343.15 K. The dark circles are 38 mental data by Wen et al..38 experimental data by Wen et al..

energy parameter values are gCH3-CH3 = 2980.016 K, gCH2-CH3 = -2493.92 K, gC-CH3 = -5486.497 K, gCH3-CH2 = 1512.1428 K, gCH2-CH2 = -6960.18 K, and gC-CH2 = -5219.42 K. Selecting molar volumes at different temperatures for pure component i, δεii can be calculated from eq. (28). The pair interaction energy parameter values are obtained by fitting the experimental data at 298.15 K. Figure 2 shows described solvent activities of poly(sty- lene) (PS, Mw = 10,920)/toluene systems at 321.65 K. The solid line is calculated by this work. Dark circles are experi- mental data reported by Wen et al..38 The pair interaction energy parameter values are gCH2-CH3 = -2493.92 K, gCH-CH3 = -1761.21 K, gC6H5-CH3 = -586.017 K, gCH2-C6H5 = 769.864 K, gCH-C6H5 = -3017.28 K, and gC6H5-C6H5 = 194.429 K. As shown in this figure, the calculated curve agrees very well with Figure 4. Solvent activities for the PS (polystyrene, Mw=1,460)/ experimental data. water system at 273.15 K. The dark circles are experimental data Figure 3 shows described solvent activities of poly(ethyl- by Wen et al..38 ene oxide)(PEO, Mw = 100,000)/benzene system at 343.15 K. The solid line is predicted by this work. Dark circles are oxide) (PEO, Mw = 1,460)/water system at 273.15 K. The experimental data reported by Wen et al..38 The pair interac- solid line is calculated by this work. Open circles are exper- 38 tion energy parameter values are gCH2-C6H5 = 769.864 K, gO-C6H5 imental data reported by Wen et al.. The pair interaction energy = 2845.58 K, gCH2-H = -5776.11 K, and gO-H = 1494.65 K. parameter values are gCH2-H = -773.512 K, gO-H = -1635.89 K, Figure 4 shows predicted solvent activities of poly(ethylene gCH2-OH = -1598.88 K, and gO-OH = -1190.52 K. In this case,

Macromol. Res., Vol. 16, No. 4, 2008 325 S. S. Lee et al.

Figure 6. Repeat unit structures structures of perfluorinated pro- ton exchange membrane with sulfonic acid functional groups. (a) Nafion (x = 5-11) and (b) Aciplex and Flemion (m=0 or 1, n=2- Figure 5. Solvent activities for the PE (polyethylene, Mw=80,000)/ 5, x=1.5-14). benzene system at 273.15 K. The dark circles are experimental data by Wen et al..38 agreement with experimental data shows slight deviation compared with other results. The proposed model does not consider the various polydispersity of chain molecules. Par- ticularly, poly(ethylene oxide) has a high polydispersity number (Mw/Mn = 1.6), this model implicitly assumes that all the polymers are monodisperse. It is likely that this defi- ciency is responsible for the observed discrepancy between the proposed model and experimental data. Figure 5 shows described solvent activities of poly(ethyl- ene) (PE, Mw = 80,000)/benzene system at 273.15 K. The solid line is predicted by this work. This result is predicted by using only previously obtained pair interaction energy parameters. Open circles are experimental data reported by Wen et al..38 The pair interaction energy parameter values are gCH2-C6H5 = 769.864 K, and gCH2-H = -5776.11 K. These Figure 7. Solvent activities for the Nafion (E.I. DuPont de Nemours parameters are not obtained by fitting experimental data. and Company)/water system at 353.15 K. Nafion 125 (EW=1,200). 22 Perfluoro-Sufonated Polymer Electrolyte Membranes/ Dark circles are experimental data by Hinatsu et al.. Water System. We propose the gE group-contribution model to predict water activities of perfluorosulfonic acid polymer electrolyte systems. We employ the chain length dependence Table III. Middle-Range Interaction Parameters; bij, cij of the high molecular weight distribution to extend the pre- ijbij cij 19 − vious model to polymer/solvent systems. Our proposed H2OSO3 0.8 -3.8 model has two kinds of adjustable model parameters: mid- + H2OH 0.8 -3.8 dle-range parameters (bij, cij) and short-range parameter (gij). − + Figure 6 shows basic structures of Nafion, Aciplex and SO3 H 1.1 18.2 Flemion. Figure 7 compares calculated water activities of Nafion 125 (duPont, EW=1,200) with experimental data 35 reported by Hinatsu et al.. Model interaction parameter Table IV. Short-Range Interaction Parameters; gij − + − + values are bH2O-SO3 = 0.818, bH2O-H = 0.818, bSO3 -H = 1.131, ijgij − + − + cH2O-SO3 = -3.768, cH2O-H = -3.768, cSO3 -H = 18.162, gCF-H2O CF 4470.8 = 4470.82, gCF2-H2O = 344.832, gCF3-H2O = 3866.79, gO-H2O = 7947.35. These parameter values are obtained using only CF2 1344.8 Nafion125/water system. As shown in this figure, the calcu- H2O CF3 3866.8 lated curve agrees well with experimental data. All parame- O 7947.4 ters are listed in Tables III and IV.

326 Macromol. Res., Vol. 16, No. 4, 2008 Calculation of Activities of Polymer Solutions Using Group Contribution Method Based on Modified Double Lattice Model

Figure 8. Solvent activities for the Nafion (E.I. DuPont de Nemours Figure 10. Solvent activities for the Aciplex(Asahi Chemical and Company)/water system at 353.15 K. Nafion 117 (EW=1,100). Industry Co.)/water system at 353.15 K. Aciplex 12 (EW=1,080). Dark circles are experimental data by Hinatsu et al..22 Dark circles are experimental data by Hinatsu et al..22

mon non-electrolyte polymer/solvent systems and much more experimental data is needed in order to calculate group parameters of electrolyte polymer/solvent systems. We con- sidered polymer solutions at temperature well below the solvent’s critical temperature. We expect that the free vol- ume effect as described by Patterson39 is almost negligible in our proposed model systems. Also, various flexibilities of chain molecules and polydispersities are not considered in this study. The model implicitly assumes that polymer has the same flexibility and monodispersity. Furthermore, sol- vent molecules are considered to be monomers where the concept of flexibility does not apply.

Conclusions Figure 9. Solvent activities for the Flemion (Asahi Glass)/water system at 353.15 K. Flemion4 (EW=890). Dark circles are exper- We proposed a group-contribution model based on a mod- et al 22 imental data by Hinatsu .. ified double lattice theory. The proposed model includes the long range interaction and middle range interaction from Figures 8, 9 and 10 represent water activities of and charge effects and the short-range interaction is covered Nafion 117 (duPont, EW=1,100), Flemion (Asahai Glass, form modified double lattice model. The proposed model EW=890) and Aciplex (Asahai Chemical, EW=1,080). Solid has a simplified and improved expression for the Helmholtz lines are calculated by using previously obtained model energy of mixing for polymer/solvent systems that includes interaction parameters with no additional adjustable model the combinatorial entropy contribution, the van der Waals parameters. Dark circles are experimental data reported by energy contribution and the specific energy contribution. Hinatsu et al..35 Pair interaction parameter which is calculated by group con- The ultimate goal of the group-contribution model lies in tribution method helps to understand what type of pair its ability to predict physical properties for systems which group parameter influences experimental results. We showed are not included in the experimental data, that is, the set of several solvent activities of various binary polymer/solvent data uses to determine the parameters. The proposed model systems comparing calculated results with experimental agrees still very well with the experimental data using pre- data. The proposed model gives very good agreement with viously obtained parameters. experimental data. In this study, we determined model interaction parameters between polymer and solvent groups. Acknowledgements. This work is supported in part by More group parameters are required to cover most com- the Ministry of Information & Communication of Korea

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