The Chemical Thermodynamic Models for Calculating the Solvent

Article pubs.acs.org/jced

The Chemical Thermodynamic Models for Calculating the Solvent Activity Coefficient of Semidiluted Aqueous and Nonaqueous Polymer Solutions in Vapor−Liquid Equilibrium Rahmat Sadeghi,† Yasaman Shahebrahimi,*,‡,§ and Ashkan Zonnouri§ †

Department of Chemistry, University of Kurdistan, Sanandaj, Iran Department of Chemical Engineering, Arak University, Arak, Iran § Department of Chemical and Petroleum Engineering, Masjed Soleiman Branch, Islamic Azad University, Masjed-Soleiman, Iran ‡

ABSTRACT: New combinational models were proposed to predict the phase behavior of vapor−liquid equilibrium of some semidiluted binary polymer solutions at different temperatures. On the basis of these models in a polymer solution a chemical reaction, solvation, was considered producing a complex composition. In addition, the Flory−Huggins theory was applied to account for the difference of molecular volumes, and two segment-based local composition models, NRTL (nonrandom twoliquid) and Wilson, were considered for description of physical interactions between species in solutions. To evaluate the accuracy of these suggested models, a number of experimental data were correlated. It was shown that the results of the proposed models are comparable with experimental data and can predict solvent activity more accurately than the available models. Gibbs energy (Gex) models and the equation of state (EoS) models. Principal Gex models for phase equilibrium calculations of polymer solution are those of Flory and Huggins,7,8 who developed an expression based on lattice theory to describe the nonidealities of polymer solutions and of Edmond and Ogston,9 who modeled nonidealities with a truncated osmotic virial expansion based on McMillan−Mayer theory.10 Other types of Gex models are local composition models such as UNIQUAC (universal quasi-chemical), UNIFAC (UNIQUAC functional group activity coefficient), NRTL (nonrandom two-liquid), NRF (nonrandom factor) and Wilson that have been used to describe the thermodynamics of polymer solutions.11−15 Some researchers modified these models to improve their accuracy. Zhao et al. extended the Wilson model for electrolyte solutions.16 Wu et al. developed the modified NRTL model for the representation of the Helmholtz energy of polymer solutions.17 Sadeghi outstretched segment-based Wilson and NRTL models for the excess molar enthalpies of a polymer solution and

1. INTRODUCTION Understanding the phase behavior of a polymer solution is useful in manufacturing steps such as polymerization and devolatilization, processing, and material development (for the advanced polymeric materials designing), polymeric membrane fabrication, the analysis of the swelling behavior of polymeric membranes, reasonable design of typical separation such as distillation, recovery of organic vapors from waste-air streams using a polymeric membrane, pervaporation, optimum formulation of paints and coatings and functionalization of nanoparticles in aqueous media.1−6 Laboratory determination of these data might be expensive and time-consuming. Therefore, using thermodynamics is a main tool for extending experimental data that have been obtained under limited conditions. Thermodynamics provides a data generation procedure at other conditions and reduces costs and saves time. For investigating the phase behavior of polymer solutions, knowing the information about solvent activity is very important. So it is necessary to predict the solvent activity in polymer solutions using the activity model equations. There are two categories of models for the description of thermodynamic properties of polymer solutions; the excess © XXXX American Chemical Society

Received: April 22, 2015 Accepted: August 7, 2015

A

DOI: 10.1021/acs.jced.5b00362 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

where γk is an activity coefficient of species k, T is the absolute temperature, and R is the gas universal constant, respectively. 2.1. Chemical Contribution. In a chemical contribution, the solvation reaction between polymer and solvent was considered which causes complex compositions to generate in polymer solutions. For this type of reaction, an equilibrium constant due to the hydrogen bonds between solvent and solute, based on the following reaction, is28

extended the Wilson model to predict phase behavior of multicomponent polymer solutions.18,19 Success in the modeling process critically depends on accurate descriptions of the thermodynamics properties and phase behavior of chemical systems. Thus, an accurate model must cover each property of solutions. To accomplish this goal and improve the accuracy of the models, researchers proposed other types of models that were introduced as combinational models. For instance, Haghtalab et al. rendered the UNIQUAC−UNIFAC model to study phase behavior of two-phase aqueous polymer systems.20,21 Radfernia et al. applied UNIQUAC-NRF to calculate the vapor−liquid equilibrium of polymer solutions by using the concept of segment.22 Other researchers used Prausnitz’s concept for the UNIQUAC model and considered two contributions for proposed models, residual contribution and combinatorial contribution. In residual contribution, physical interactions between particles and, in combinatorial contribution, physical properties such as the structure and shape of particles were taken into account. Pedrosa et al. extended the UNIQUAC model for the residual term and applied the Flory−Huggins equation for free entropic volume as a combinatorial term.23 Moreover, Chen and Vetere used a modified NRTL model for the interaction between segments and the Flory−Huggins model for the entropy contribution to calculate the phase equilibrium of polymer solutions.24,25 We previously applied combinational models for the VPO (vapor-pressure osmometry) equilibrium calculations of polymer solutions using some local composition models such as NRTL, Wilson, and UNIQUAC for residual contribution, and the Flory-Huggins model for combinatorial contribution.26,27 In 1992, Yu and Nishiumi recommended a thermodynamic theory of phase separation in mixtures with lower critical solution temperature. Their theory incorporates (1) chemical equilibrium theory to account cross solvation between solute and solvent, (2) Flory−Huggins theory for the difference of molecular volumes, and (3) NRTL theory for the interaction between molecules.28 Also Yu et al. applied this concept to phase separation in the aqueous solution of polymers.29 In this study, based on Yu and Nishiumi’s theory, new combinational models have been introduced and modified for solvent activity calculations in vapor−liquid equilibrium of some semidiluted binary polymer solutions at different temperatures. Further Flory−Huggins model as combinatorial contribution and two local composition models, NRTL and Wilson, as residual contribution have been considered.

iSl + jWl ⇔ SiWj

G =

E Gchem

+

E Gphys

i

j

ϕSi ϕWj l

(5)

l

where i and j are average solvation numbers. Sl is polymer species, Wl is solvent species, and SiWj is a complex composition. ϕSl, ϕWl and ϕSiWj are the true volume fractions of the molecular species. Sl, Wl and SiWj, are, respectively. The true volume fraction can be obtained by simultaneously solving eq 5 and the following material balance equations: ϕS = ϕS + ϕS W l

i

j

i i + jη

ϕW = ϕW + ϕS W l

i

j

(6)

jη i + jη

(7)

where ϕS and ϕW are the apparent volume fraction of chemical species and η is the ratio of molecular volume of solvent to solute. From Flory−Huggins theory, the excess Gibbs free energy GE can be expressed as follow:4 mixt

mixt

g g GE = ntreal real − ntideal ideal RT RT RT

(8)

where mixt greal

RT

= 1Sl ln ϕS + 1Wl ln ϕW + 1SiWj ln ϕS W l

mixt g ideal

RT

1klνk + 1SiWj 1SlνSl + 1WW l l

ntreal ntreal

1SiWj =

(2)

j

(9)

(10)

(k = Sl , Wl , SiWj) (11)

(12)

nWl − jnSiWj

1Wl =

xk =

i

nSl − inSiWj

1Sl =

(1)

l

= xS ln xS + xW ln xW

ϕk =

Also for species k, it can be written:

1 ∂GE = ln γk RT ∂nk

ϕS W

k=

2. OUTLINE OF THE THERMODYNAMICS MODEL Consider a solution of NW solvent molecules and NS solute molecules at absolute temperature T and pressure P. The excess Gibbs free energy GE, as consisting of two additives parts GEchem (chemical contribution) and GEphys (physical contribution), was proposed as follow:28 E

(4)

(13)

nSiWj ntreal

(14)

nk ntideal

(15)

where g is molar Gibbs free energy in real and ideal solution, νk is molar volume, nk is the number of moles, 1k is the true and xk is the apparent mole fraction of species k. In addition, nt is the total mixt

So ln γk = (ln γk)chem + (ln γk)phys

(3) B



Article

Table 1. Parameters of the NRTL + Chemical Model and Wilson + Chemical Model for Data of Binary Aqueous Polymer Solutions23 at T = 298.15 K NRTL + Chemical model

Wilson + Chemical model

τ12

τ21

E12

E21

system

j

k

J mol−1

J mol−1

j

k

J mol−1

J mol−1

water(1) + PEGDME250(2) water(1) + PEGDME500(2) water(1) + PEGDME2000(2) water(1) + PEGMME350(2) water(1) + PEG200(2) water(1) + PEG6000(2) water(1) + PPG400(2)

11.2512 22.5022 90.0081 15.4201 8.3253 249.7491 18.0843

1.0176 5.6917 1.26999 7.4558 1.2472 1.2369 2.7267

0.9929 0.9732 0.9769 0.9618 0.9635 0.9987 0.9610

−0.1080 −0.3373 −0.6206 −0.2058 −0.0149 −0.8034 −0.1051

11.2513 22.5024 46.6113 18.847 6.6445 286.6435 18.08431

1.6438 7.5840 14.7314 4.02001 2.9426 2.7199 5.1739

21.4382 45.1697 41.2717 52.8256 99.9156 10.7268 80.8253

−11058.662 −8745.212 −998.444 −11108.013 −10000.001 −981.453 −11109.836



τ12 system

j

k


11.2512 22.5022 90.0081 15.4201 8.3251 249.7491 18.0843

1.49728 7.2070 1.2699 7.4561 1.2814 1.2369 2.7269

J mol

τ21 −1

E12 −1

j

k

11.2513 22.5024 46.6114 18.8475 6.6445 286.6434 18.0843

2.2099 7.5840 15.2014 18.7669 3.220 2.8096 5.3489

J mol

−0.0141 −0.2847 −0.6160 −0.1319 −0.1535 −0.7593 −0.1252

0.9847 0.9774 0.9844 0.9648 1.0232 0.9998 0.9751

E21 −1

J mol−1

J mol

32.5243 45.1697 26.1227 73.5918 21.0885 10.1255 57.6352

−1250.339 −8745.212 −998.769 −12437.171 −9998.178 −1905.611 −11110.509



τ12

τ21

E12

E21

system

j

k

J mol−1

J mol−1

j

k

J mol−1

J mol−1


11.2512 22.5022 90.0081 15.4201 8.3251 249.7491 18.0843

1.7226 7.4602 1.2700 7.4562 1.3018 1.2370 2.7271

1.0066 0.9764 0.9945 0.9704 1.0077 1.0035 0.9804

−0.1306 −0.2475 −0.6268 −0.1718 −0.1446 −0.7797 −0.1321

11.2513 22.5024 46.6114 18.8475 6.6445 286.6434 18.0843

8.0346 15.6547 15.7507 22.5563 3.9017 3.0048 5.5143

44.9190 56.3653 11.2047 68.1492 48.5542 11.4967 46.7001

−12249.256 −9608.827 −999.0699 −12437.308 −9997.834 −1905.575 −1111.135

mixt g id Gideal = nt ideal RT RT

mole of components in real and ideal solutions. In real s, there are polymer, solvent, and a new component as a complex. But in ideal solutions the chemical reaction does not occur and two ideal typical components, solvent and solute, exist. Thus, nreal t and nt are defined as follow: ntreal

= nS + nW + (1 − i − j)nSiWj

= nS ln xS + nW ln xW ⎛ nS ⎞ ⎛ nW ⎞ = nS ln⎜ ⎟ + nW ln⎜ ⎟ ⎝ nS + nW ⎠ ⎝ nS + nW ⎠

(16)

ntideal = nS + nW

By substituting eqs 18 and 19 in eq 8 and using eq 2: ln γWchem = −ϕW + 1 + ln ϕW − ηϕS

(17)

l

i

i

j

j

vWl vSiWj

l

⎛ nW ⎞ − ln⎜ ⎟ ⎝ nW + nS ⎠

(20)

As mentioned SiWj is a complex composition and its molar volume (vSiWj) cannot be obtained. Thus, for calculating vWl/(vsiWj) in eq 20, the following equations can be used:

mixt

mixt g Greal = ntreal real = (nSl − inSiWj) ln ϕS l RT RT l

l

− ϕS W

The Gibbs free energy for real and ideal solution can be calculated by

+ (nWl − jnSiWj) ln ϕW + nSiWj ln ϕS W

(19)

vWl = ηvSl

(18) C

(21) DOI: 10.1021/acs.jced.5b00362 J. Chem. Eng. Data XXXX, XXX, XXX−XXX


Article

Table 4. Parameters of the NRTL + Chemical Model and Wilson + Chemical Model for Data of Binary Aqueous Polymer Solutions24 at T = 318.15 K NRTL+ Chemical model


τ12

τ21

E12

E21

system

j

k

J mol−1

J mol−1

j

k

J mol−1

J mol−1

water(1) + PEGDME250(2) water(1) + PEGDME500(2) water(1) + PEGDME2000(2) water(1) + PEG400(2)

11.2512 22.5022 90.0081 16.6500

2.1459 8.2752 1.3781 1.5179

0.9801 0.9602 0.9743 0.9786

0.0176 −0.1650 −0.5252 −0.7797

11.2513 22.5022 46.6114 14.5023

9.5827 23.0665 15.8018 2.4123

72.8159 86.4579 35.8843 47.2786

−13463.411 −10860.275 −2384.512 −10000.002

Table 5. Parameters of the NRTL + Chemical Model and Wilson + Chemical Model for Different Methanol (1) + Polymer (2) Systems24 at T = 318.15 K NRTL+ Chemical model τ12 −1

system

j

k

methanol(1) + PEGDME250(2) methanol(1) + PEGDME500(2) methanol(1) + PEGDME2000(2) methanol(1) + PEG400(2) methanol(1) + PPG400(2) methanol(1) + PEGDME250(2)

4.9902 9.9803 39.9235 7.3853 8.0231 4.9902

1.1080 1.0857 0.6993 2.0076 0.8199 1.1080

Wilson + Chemical model τ21

E12 −1

J mol

J mol

0.9730 0.9545 0.9378 0.9461 1.11768 0.9730

0.2342 −0.0784 −0.4461 −0.0042 0.9572 0.2342

j

k

2.8673 9.1189 45.8153 6.9114 7.5885 45.8153

0.8410 0.9962 0.9108 0.3888 0.2048 0.9108

E21 −1

J mol

44.5542 85.0164 133.6084 44.6593 44.6145 133.6084

J mol−1 −11054.621 −11054.524 −7925.516 −11054.622 −11054.624 −7925.516

Table 6. Parameters of the NRTL + Chemical Model and Wilson + Chemical Model for Different Ethanol (1) + Polymer (2) Systems24 at T = 318.15 K NRTL + Chemical model


τ12

τ21

E12

E21

system

j

k

J mol−1

J mol−1

j

k

J mol−1

J mol−1

ethanol(1) + PEGDME250(2) ethanol(1) + PEGDME500(2) ethanol(1) + PPG400 (2) ethanol (1) + PEG400(2)

3.464 6.9286 5.567 5.126

7.3518 7.351 6.9960 6.9932

0.9809 0.9435 0.9671 0.9549

1.0005 0.0492 −0.1810 0.07385

2.6235 5.4513 5.7193 4.1013

8.1168 1.5773 4.7482 3.3002

44.0571 45.6882 43.9443 44.3668

−11053.619 −11054.622 −11054.999 −11054.621

Table 7. Parameters of the NRTL + Chemical Model and Wilson + Chemical Model for Different 2-Propanol (1) + Polymer (2) Systems24 at T = 318.15 K NRTL+ Chemical model


τ12

τ21

E12

E21

system

j

k

J mol−1

J mol−1

j

k

J mol−1

J mol−1

2-propanol(1) + PEGDME250(2) 2-propanol(1) + PEGDME500(2) 2-propanol(1) + PPG400 (2) 2-propanol(1) + PEG400(2)

2.6464 5.282 4.244 3.908

7.9622 7.97100 7.9790 7.3531

0.9711 0.9212 0.9480 0.9494

0.7912 0.4787 0.1476 0.5180

1.5682 1.8263 3.0271 1.473

27.5752 1.4062 5.8985 3.0524

42.2013 44.5183 86.7141 44.4032

−11054.595 −11054.624 −11095.914 −11054.612

ϕSs, ϕWl, and ϕSiWj must be identified. To obtain these values and simplify the model, two assumptions have been considered: (1) i = 1 and (2) concentration of Sl ≪ concentration of Wl and Si Wj. The first assumption refers to one molecule of solute (polymer), which has been surrounded by a number of solvent molecules in solution. The second assumption holds that in a semidiluted polymer solution, most polymer molecules take part in solvation reactions and cannot be considered separately in solution. Therefore, by using these assumptions, ϕSiWj in eq 6 can be rewritten as follows:

and vSiWj = (i + jη)vSl

(22)

thus vWl vSiWj

=

vWl vSl

×

vSl vSiWj

=η×

1 i + jη

(23)

Therefore, the chemical solvent activity coefficient can be expressed as ln γWChem = ln ϕW + 1 − ϕW − ηϕS − l

l

l

ϕS W η i

j

i + jη

− ln xW

ϕS W = ϕS(1 + jη)

(24)

i

D

j

(25) DOI: 10.1021/acs.jced.5b00362 J. Chem. Eng. Data XXXX, XXX, XXX−XXX


Article

Table 8. Comparison of Absolute Relative Deviations (ARD) of Solvent Activity between Proposed Models and Old Models ARD % In previous works

ARD % in this work

system

T/K

ref

NRTL + F−H

Wilson + F−H

UNIQUAC + F−H

water(1) + PEGDME250(2) water(1) + PEGDME500(2) water(1) + PEGDME2000(2) water(1) + PEGMME350(2) water(1) + PEG200(2) water(1) + PEG6000(2) water(1)+PPG400(2) water(1) + PEGDME250(2) water(1) + PEGDME500(2) water(1) + PEGDME2000(2) water(1) + PEGMME350(2) water(1) + PEG200(2) water(1) + PEG6000(2) water(1) + PPG400(2) water(1) + PEGDME250(2) water(1)+PEGDME500(2) water(1)+PEGDME2000(2) water(1) + PEGMME350(2) water(1) + PEG200(2) water(1) + PEG6000(2) water(1) + PPG400(2) water(1) + PEGDME250(2) water(1) + PEGDME500(2) water(1) + PEGDME2000(2) water(1) + PEG400(2) methanol(1) + PEGDME250(2) methanol(1) + PEGDME500(2) methanol(1) + PPG400(2) methanol(1)+PEG400(2) ethanol(1) + PEGDME250(2) ethanol(1) + PEGDME500(2) ethanol(1) + PEG400(2) ethanol(1) + PPG400(2) 2-propanol(1) + PEGDME250(2) 2-propanol(1) + PEGDME500(2) 2-propanol(1) + PEG400(2) 2-propanol(1) + PPG400(2)

298.15 298.15 298.15 298.15 298.15 298.15 298.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 308.15 308.15 308.15 308.15 308.15 308.15 308.15 318.15 318.15 318.15 318.15 318.15 318.15 318.15 318.15 318.15 318.15 318.15 318.15 318.15 318.15 318.15 318.15

23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24

0.10 0.01 0.06 0.02 0.16 0.02 0.01 0.08 0.01 0.06 0.01 0.05 0.01 0.02 0.09 0.01 0.06 0.02 0.03 0.01 0.02 0.10 0.06 0.12 0.02 0.04 0.05 0.1 0.04 0.05 0.05 0.03 0.03 0.08 0.09 0.02 0.02

0.1 0.01 0.05 0.02 0.16 0.02 0.03 0.06 0.01 0.07 0.01 0.05 0.01 0.01 0.09 0.01 0.06 0.02 0.03 0.01 0.02 0.11 0.08 0.09 0.02 0.03 0.06 0.09 0.04 0.05 0.05 0.03 0.05 0.05 0.05 0.02 0.02

0.09 0.02 0.05 0.02 0.17 0.02 0.03 0.08 0.01 0.05 0.01 0.05 0.01 0.02 0.08 0.01 0.05 0.02 0.03 0.02 0.01 0.09 0.08 0.08 0.01

By substituting eq 25 in eq 7 the following is obtained: ϕW = ϕW − jηϕS

ln γWNRTL =

(26)

l

Moreover, using the above equations and substituting eq 5 in eq 6 we have ϕS = l

S k ϕ ( W 1 + jη

− jηϕS) j

+

ϕη

1+

− jηϕS)

j

0.021 0.014 0.015 0.017 0.025 0.017 0.006 0.017 0.010 0.014 0.005 0.028 0.013 0.013 0.021 0.007 0.016 0.013 0.022 0.017 0.003 0.041 0.042 0.027 0.020 0.016 0.009 0.006 0.016 0.021 0.017 0.023 0.035 0.031 0.038 0.015 0.01

⎞ ⎛ XS exp(− ατSW ) 1 − XS ⎟ ⎜− 1 − XS + XS exp(− ατSW ) ⎝ 1 − XS + XS exp(− ατSW ) ⎠

where

XS =

ln γWChem = ln(ϕW − jηϕS) + 1 − (ϕW − jηϕS) S k ( ϕ W 1 + jη

0.018 0.010 0.020 0.021 0.025 0.022 0.006 0.021 0.005 0.025 0.02 0.019 0.008 0.012 0.012 0.009 0.005 0.020 0.018 0.01 0.007 0.04 0.051 0.035 0.020 0.016 0.009 0.009 0.017 0.021 0.018 0.019 0.034 0.016 0.05 0.012 0.017

(29)

(27)

Hence, based on chemical model, the activity coefficient of solvent can be expressed as

−


XS exp(− ατSW )τSW XS exp(− ατWS) + (1 − XS) + XS exp(− ατSW ) (1 − XS) exp(− ατWS) + XS ⎛ (1 − XS) exp(− ατWS)τWS ⎞ × ⎜τWS − ⎟ XS + (1 − XS) exp(− ατWS) ⎠ ⎝

ϕ

1+

NRTL + Chemical model

r=

− ηϕS − ln xW (28)

2.2. Physical Contribution. The physical contribution for activity coefficient of solvent is represented by two segmentbased local composition models, the nonrandom two-liquid (NRTL) model and the Wilson model, as follow:24,19

xSr 1 − xS + xSr

1 η

τmn =

(30)

(31)

amn RT

(32)

and E



Article

⎛ ⎜ ⎛ ⎛ − E ⎞⎞ ln = CrW ⎜ln⎜1 − XS + XS exp⎜ SW ⎟⎟ ⎝ CRT ⎠⎠ ⎜ ⎝ ⎝ ⎛ ESW ⎜ 1 − 1 − XS + XS exp − CRT + (1 − XS)⎜ ESW ⎜ 1 − XS + XS exp − CRT ⎝

3. RESULTS AND DISCUSSION In this study, two combinational models have been proposed to predict solvent activity coefficient, γw, in some semidiluted aqueous and nonaqueous polymer solutions at different temperatures. Also, the definition of ϕs was modified on the basis of Belfiore et al.:30

γWWilson

(

)) ⎞⎟

(

(

)

⎟ ⎟ ⎠

⎛ EWS EWS ⎜ exp − CRT − XS + (1 − XS) exp − CRT + XS⎜ EWS ⎜ XS + (1 − XS) exp − CRT ⎝

) (

(

( )

(

φS =

⎞

)) ⎞⎟⎟ ⎟⎟ ⎟⎟ ⎠⎠

ln γw = ln γwchem + ln γwNRTL = nγWChem = ln(ϕW − jηϕS) + 1 − (ϕW − jηϕS) ϕη

1+

S k (ϕW 1 + jη

− jηϕS) j

− ηϕS − ln xW

+

XS exp(− ατSW )τSW (1 − XS) + XS exp(− ατSW )

+

XS exp(− ατWS) (1 − XS) exp(− ατWS) + XS

⎛ (1 − XS) exp(− ατWS)τWS ⎞ ×⎜τWS − ⎟ XS + (1 − XS) exp(− ατWS) ⎠ ⎝ 1 − XS + 1 − XS + XS exp(− ατSW ) ⎛ ⎞ XS exp(− ατSW ) ×⎜− ⎟ ⎝ 1 − XS + XS exp(− ατSW ) ⎠

(34)

and ln γw = ln γwchem + ln γwWilson = ln γWChem = ln(ϕW − jηϕS) φSη + 1 − (ϕW − jηϕS) − − ηϕS k 1 + 1 + jη (φW − jηφS) j − ln xW

⎛ ⎜ ⎛ ⎛ − E ⎞⎞ + CrW ⎜ln⎜1 − XS + XS exp⎜ SW ⎟⎟ ⎝ CRT ⎠⎠ ⎜ ⎝ ⎝

⎛ ESW ⎜ 1 − 1 − XS + XS exp − CRT + (1 − XS)⎜ ESW ⎜ 1 − XS + XS exp − CRT ⎝

(

(

(

)

)) ⎞⎟ ⎟ ⎟ ⎠

⎛ EWS EWS ⎜ exp − CRT − XS + (1 − XS) exp − CRT + XS⎜ EWS ⎜ XS + (1 − XS) exp − CRT ⎝

(

) (

(

( )

))

(1 − xS) + xzr 0.75

(36)

As can be seen, these proposed models have four adjustment parameters: j refers to solvation number, k is equilibrium constant, and the other two parameters refer to physical interactions between species in solution. The investigated systems are some binary aqueous and nonaqueous solutions of poly(ethylene glycol) dimethyl ether 250 (PEGDME 250), poly(ethylene glycol) dimethyl ether 500 (PEGDME 500), poly(ethylene glycol) dimethyl ether 2000 (PEGDME 2000), poly(ethylene glycol) monomethyl ether 350 (PEGMME 350), poly(ethylene glycol) 200 (PEG 200), poly(ethylene glycol) 6000 (PEG 6000) and poly(propylene glycol) 400 (PPG 400) in water, methanol, ethanol and 2-propanol at 298.15 K, 303.15 K, T = 308.15 K and T = 318.15 K.26,27 For the application of new models the value of r = 1 was used for solvents, and for a polymer species the value of r was considered as the ratio of the molar volume of polymer to that of the solvent at 298.15 K. The molar volumes of polymers have been calculated from the specific volumes and the number the average amount of molar masses of polymers. The specific volume of PEGDME, PEG, and PPG at 298.15 K are 0.9663,26 0.8387,31 and 0.9962,32 respectively. The values of nonrandomness factor, α, and the effective coordination number C were set to 0.2 and 10.24,33 Tables 1 to 7 show the obtained parameters of these new models for the studied polymer systems in some solvents at different temperatures, and in Table 8 the obtained solvent activity in aqueous and nonaqueous polymer solutions from these new proposed models based on Absolute Relative Deviation (ARD) were compared with some available models. It can be seen that because of having the least ARD, new combinational models can predict phase behavior of polymer solution more accurately than the available models. In Figure 1 the calculated solvent activity from new thermodynamics model (Chemical + NRTL) of some binary aqueous solutions such as water(1) + PEGDME250(2), water(1) + PEGDME2000(2) at 298.15 K, 308.15 K, and 318.15 K have been compared with experimental data. Also Figure 2 shows the experimental data and obtained solvent activity from the new thermodynamics model (Chemical + Wilson) of some binary nonaqueous solutions such as ethanol(1) + PEGDME250(2), ethanol(1) + PEG400(2), and ethanol(1) + PPG400 at 318.15 K. As can be seen from Tables 1 to 7, the solvation number of a specific polymer solution, j, increases with increasing molecular weight of the polymer. For a polymer solution, the equilibrium constant, k, also increases with increasing temperature.

(33)

Where α is nonrandomness factor in the range of 0.2 to 0.3 and has no significant effect on the behavior of the model,2 X is the mole fraction of segment, C is the effective coordination number in the system.19 A value of rw = 1 was used for solvents, τmn and Emn are adjustable parameters in NRTL and Wilson models and refer to the interaction between components n and m in solution. 2.3. The Proposed Model. The combination of chemical model eq 28) with NRTL model (eq 29) or Wilson model (eq 33 leads to the final equations of proposed models as follow:

−

xSr 0.75

4. CONCLUSIONS New combinational models that considered solvation reaction and physical interactions have been utilized in some semidiluted aqueous and nonaqueous polymer solutions in order to correlate the experimental solvent activity data at different temperatures. For the residual term that infers

⎞⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎠⎠ (35) F



Article

Figure 1. Plot of the experimental water activity and that obtained from the Chemical + NRTL model against the mass fraction of polymer for water(1) + PEGDME250(2) systems and water(1) + PEGDME2000(2) systems at different temperatures.

Figure 2. Plot of the experimental and obtained solvent activity of ethanol from the Chemical + Wilson model against the mass fraction of polymer in ethanol(1) + PEGDME250(2), ethanol(1) + PEG400(2) and ethanol(1) + PPG400(2) systems at T = 318.15 K.

physical interactions between particles in solution, two segment-based local composition models, NRTL and Wilson models, have been applied. In each polymer solution, the increasing molecular weight of polymer causes a raise in the solvation number. Also increasing the temperature leads to an increase in equilibrium constant. It was shown that the proposed models could correlate the experimental data of solvent activity more accurately than available models.

Therefore, it seems that solvation reaction assumed in the polymer solutions is true.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. G



■

Article

(24) Chen, C. C. A Segment-based Local Composition Model for the Gibbs Energy of Polymer Solutions. Fluid Phase Equilib. 1993, 83, 301− 312. (25) Vetere, A. Rules for Predicting Vapor-Liquid Equilibria of Amorphous Polymer Solutions Using a Modified Flory-Huggins Equation. Fluid Phase Equilib. 1994, 97, 43−45. (26) Sadeghi, R.; Shahebrahimi, Y. Vapor-Liquid Equilibria of aqueous polymer solutions from Vapor-Pressure Osmometry and Isopiestic measurements. J. Chem. Eng. Data 2011, 56, 789−799. (27) Sadeghi, R.; Shahebrahimi, Y. Vapor Pressure Osmometry Determination of Solvent Activities of Different Aqueous and Nonaqueous Polymer Solutions at 318.15 K. J. Chem. Eng. Data 2011, 56, 2946−2954. (28) Yu, M.; Nishiumi, H. Theory of Phase Separation in Mixtures with Lower Critical Solution Temperature. J. Phys. Chem. 1992, 96, 842−845. (29) Yu, M.; Nishiumi, H.; Arons, J. d. S. Thermodynamics of Phase Separation in Aqueous Solutions of Polymers. Fluid Phase Equilib. 1993, 83, 357−364. (30) Belfiore, L. A.; Patwardhan, A. A.; Lenz, T. G. Shortcomings of UNIFAC-FV to Characterize the Phase Behavior of Polymer-Polymer Blends. Ind. Eng. Chem. Res. 1988, 27, 284−294. (31) Sadeghi, R.; Hosseini, R.; Jamehbozorg, B. Effect of Sodium Phosphate Salts on the Thermodynamic Properties of Aqueous Solutions of Poly (ethylene oxide) 6000 at Different Temperatures. J. Chem. Thermodyn. 2008, 40, 1364−1377. (32) Sadeghi, R.; Jamehbozorg, B. Volumetric and Viscosity Studies of Interactions Between Sodium Phosphate Salts and Poly(propylene glycol) 400 in Aqueous Solutions at Different Temperatures. Fluid Phase Equilib. 2009, 284, 86−98. (33) Zhao, E.; Yu, M.; Sauve, R. E.; Khoshkbarchi, M. K. Extension of the Wilson Model to Electrolyte Solutions. Fluid Phase Equilib. 2000, 173, 161−175.

REFERENCES

(1) High, M. S.; Danner, R. P. Prediction of Solvent Activities in Polymer Solutions. Fluid Phase Equilib. 1990, 55, 1−15. (2) Ballantine, D. S.; Wohltjen, H. Surface Acoustic Wave Device for Chemical Analysis. Anal. Chem. 1989, 61, 704−715. (3) Grate, J. W.; Klusty, M.; McGill, R. A.; Abraham, M. H.; Whiting, G.; Andonian- Haftvan, J. The Predominate Role of Swelling − Induced Modulus Changes of the Sorbent Phase in Determining the Responses of Polymer− Coated Surface Acoustic Wave Vapour Sensor. Anal. Chem. 1992, 64, 610−624. (4) Sadeghi, R.; Shahebrahimi, Y. Vapor−Liquid Equilibria of Aqueous Polymer Solutions from Vapor-Pressure Osmometry and Isopiestic Measurements. J. Chem. Eng. Data 2011, 56, 789−799. (5) De Lisi, R.; Gradzielski, M.; Lazzara, G.; Milioto, S.; Muratore, N.; Prevost, S. Aqueous Laponite Clay Dispersions in the Presence of Poly(ethylene oxide) or Poly(propylene oxide) Oligomers and their Triblock Copolymers. J. Phys. Chem. B 2008, 112, 9328−9326. (6) Maldiney, T; Richard, S.; Seguin, J.; Wattier, N.; Bessodes, M.; Scherman, D. Effect of Core Diameter, Surface Coating, and PEG Chain Length on the Biodistribution of Persistent Luminescence Nanoparticles in Mice. ACS Nano 2011, 5, 854−862. (7) Flory, P. J. Principles of Polymer Chemistry; Cornell University: Ithaca, 1953. (8) Huggins, M. L. Solutions of Long Chain Compounds. J. Chem. Phys. 1941, 9, 440−451. (9) Edmond, E.; Ogston, A. G. An Approach to the Study of Phase Separation in Ternary Aqueous Systems. Biochem. J. 1968, 109, 569− 576. (10) McMillan, W. G.; Mayer, J. E. The Statistical Thermodynamics of Multicomponent Systems. J. Chem. Phys. 1945, 13, 276−284. (11) Abrams, D. C.; Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems. AIChE J. 1975, 21, 116−128. (12) Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. GroupContribution Estimation of Activity Coefficients in Non-ideal Liquid Mixtures. AIChE J. 1975, 21, 1086−1099. (13) Renon, H.; Prausnitz, J. M. Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures. AIChE J. 1968, 14, 135− 144. (14) Haghtalab, A.; Vera, J. H. A Nonrandom Factor Model for the Excess Gibbs Energy of Electrolyte Solutions. AIChE J. 1988, 34, 803− 813. (15) Wilson, G. M. Vapor- Liquid equilibrium.XI. New Expression for the Excess Free Energy of Mixing. J. Am. Chem. Soc. 1964, 86, 127−130. (16) Zhao, E.; Yu, M.; Sauve, R. E.; Khoshkbarchi, M. Extension of the Wilson Model to Electrolyte Solutions. Fluid Phase Equilib. 2000, 173, 161−175. (17) Wu, Y. T.; Zhu, Z. Q.; Lin, D. Q.; Mei, L. H. A Modified NRTL Equation for the Calculation of Phase Equilibrium of Polymer Solutions. Fluid Phase Equilib. 1996, 121, 125−139. (18) Sadeghi, R. Extension of the Segment-based Wilson and NRTL Models for Correlation of Excess Molar Enthalpies of Polymer Solutions. J. Chem. Thermodyn. 2005, 37, 1013−1018. (19) Sadeghi, R. Extension of the Wilson Model to Multicomponent Polymer Solutions: Applications to Polymer-polymer Aqueous Twophase Systems. J. Chem. Thermodyn. 2005, 37, 55−60. (20) Haghtalab, A.; Asadollahi, M. A. An Excess Gibbs Energy Model to Study the Phase Behavior of Aqueous Two-phase Systems of Polyethylene Glycol+Dextran. Fluid Phase Equilib. 2000, 171, 77−90. (21) Haghtalab, A.; Mokhtarani, B. On Extension of UNIQUAC-NRF Model to Study the Phase Behavior of Aqueous Two Phase Polymer− salt Systems. Fluid Phase Equilib. 2001, 180, 139−149. (22) Radfarnia, H. R.; Bogdanic, G.; Taghikhani, V.; Ghotbi, C. The UNIQUAC-NRF Segmental Interaction Model for Vapor-Liquid Equilibrium Calculations for Polymer Solution. Polimeri 2005, 26, 115−120. (23) Pedrosa, N.; Gao, J.; Marrucho, I. M.; Coutinho, A. P. Correlation of Solvent Activities in Polymer Solutions: A Comparison of Models. Fluid Phase Equilib. 2004, 219, 129−138. H


The Chemical Thermodynamic Models for Calculating the Solvent

Recommend Documents