Comparison of equations of state based on different perturbation terms for polymer systems

Fluid Phase Equilibria 206 (2003) 127–145

Comparison of equations of state based on different perturbation terms for polymer systems Changjun Peng, Honglai Liu∗ , Ying Hu Department of Chemistry, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, China Received 14 August 2002; accepted 28 October 2002

Abstract In this work, several well-known perturbation term of equation of state are reviewed and applied to establish different equations of state for chainlike fluids based on Hu et al.’s equation of state for hard-sphere chain fluids (HSCFs). The selected perturbation terms include Song–Lambert–Prausnitz model, Cotterman–Schwarz–Prausnitz model, O’Lenick–Chiew model, Gross–Sadowski model, Guo–Wang–Lu model, Chen–Kreglewski model, and Alder–Young–Mark model. Each equation of state is characterized by three molecular parameters, namely, the number of segments per molecule, segment size, and interaction energy between segments. Consequently, we simplified these equations of state to apply conveniently to polymers. The model parameters have been obtained by fitting pVT data of polymers. A comparison is also made with each other based on experimental data. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Equation of state; Polymer; Perturbation theory; pVT relationship

1. Introduction In recent years, there has been increased interest in the development of equation of state for chainlike molecules based on off-lattice approach [1–12]. These analytic equations of state possess physically significant parameters, which capture the effects of molecular size, shape, and interaction energy. Many useful equations of state can be expressed as the sum of a reference term and a perturbation term, representing repulsive and attractive interactions, respectively. Generally, the hard-sphere chain fluids (HSCFs) is used as the reference term in most equations of state. Sadus [13], and Kim and Bae [14] made a comparison of several well-known HSCFs equations of state with computer simulation data, respectively. Based on the HSCFs, the equation of state for practice chainlike fluids can then be established by introducing a perturbation term. In various equations of state, the perturbation contribution that accounts for nonbonded ∗

Corresponding author. Tel.: +86-21-64252767; fax: +86-21-64252485. E-mail address: [email protected] (H. Liu). 0378-3812/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 2 ) 0 0 3 1 0 - 2

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attraction between molecules is treated differently. The attractive contribution in the generalized Flory dimer theory (GFD) [1] is based on a square-well (SW) chain model and derived in the context of the generalized dimer mean field theory. The SAFT equation [3] uses the power series initially fitted by Alder et al. [15] to molecular dynamics data for SW sphere, and subsequently modified by Chen and Kreglewski [16] to model attraction between segments. The other treatment approaches of the perturbation term were adopted in different versions of the SAFT equation of state [17–20]. In the PHSC equation of state [4], the perturbation term uses a simple van der Waals type one based on the Song and Mason model [21]. In new PHSC equation of state, Hino and Prausnitz [22] adopted a perturbation type based on the analytic solution by Chang and Sandler [23,24] to the second-order perturbation theory of Barker and Henderson [25] for the SW fluid of variable width. Similar to Hino and Prausnitz’s work [22], Paredes et al. [12] established an equation of state for polymers and normal fluids using the SW potential of variable well width. Besides, Cotterman et al.’s model [26] has also been used as perturbation contribution [27]. More recently, several new expressions of perturbation term have been reported in the literature [10,28–30]. In the perturbed Lennard–Jones chain (PLJC) equation of state [9], perturbation term is determined through a first-order variational perturbation scheme [28]. The SW coordination number model (SWCNM) [29,30] was adopted by Feng and Wang [8] to develop an equation of state called PHSC-SWCNM. Gross and Sadowski [10] proposed their perturbation term by analyzing the second-order perturbation theory of Barker and Henderson [25]. Similar to Zhou and Stell [31], Hu et al. [32] derived a HSCFs equation of state through the r-particle cavity correlation function (CCF) for chains formed by sticky sphere. The r-particle CCF is approximated by a product of effective two-particle CCFs which account for nearest-neighbor and next-to-nearest-neighbor correlation. The predicted results of the final equation of state are in excellent agreement with computer simulations. Subsequently, Liu and Hu [33] established a equation of state for real chainlike fluids by using directly the Alder et al.’s SW perturbation [15]. The equation of state presented can give a better result for the correlation and calculation of pVT and VLE of pure fluids and mixtures [34]. In this work, we proposed several equations of state for chainlike fluids by combining a reference equation for HSCF based on Hu et al.’s model [32] with different attractive potential served as a perturbation term. These selected perturbation terms are Song–Lambert–Prausnitz model (SLP) [4], Cotterman–Schwarz–Prausnitz model (CSP) [26], O’Lenick–Chiew model (O’LC) [28], Gross–Sadowski model (GS) [10], Guo–Wang–Lu model (GWL) [29,30], Chen–Kreglewski model (CK) [16], and Alder– Young–Mark model (AYM) [15], respectively. Each equation of state is characterized by three molecular parameters, namely, the number of segments per molecule, segment size, and interaction energy between segments. Consequently, we simplified these equations of state to apply conveniently to polymers. The model parameters can be obtained by fitting pVT data for polymers. A comparison is also made with each other based on experimental data, rather than simulations.

2. Equation of state Following Hu et al.’s idea [32], the compressibility factor of a pure chain fluid with r segments can be written as Z=

1 + aη + bη2 − cη3 + Zpert (1 − η)3

(1)

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where a, b, c are function of chain length
 r−1r−2 r−1 a2 + a=r 1+ a3 r r r
 r−1 r−1r−2 b=r 1+ b2 + b3 r r r
 r−1 r−1r−2 c =r 1+ c2 + c3 r r r

129

(2) (3) (4)

Here a2 = 0.45696, a3 = −0.74745,

b2 = 2.10386,

c2 = 1.75503

b3 = 3.49695,

c3 = 4.83207

(5) (6)

Packing density η = (π/6)rρd 3 = (π/6)rρλ3 σ 3 . ρ is the number density of chain molecules; d the effective hard-sphere diameter of a segment (temperature-dependent); σ the hard-sphere diameter of a segment (temperature-independent); λ the dimensionless hard-sphere diameter and d = λσ. 2.1. Equation of state based on the CSP model Following Cotterman et al. [26], perturbation term of equation of state can be given by 
Z1 Z2 pert Z =r + T˜ T˜ 2

(7)

where Z1 = ρτ (−8.5959 − 2 × 4.5424ρτ − 3 × 2.1268ρτ2 + 4 × 10.285ρτ3 )

(8)

Z1 = ρτ (−1.9075 + 2 × 9.9724ρτ − 3 × 22.216ρτ2 + 4 × 15.904ρτ3 )

(9)

ρτ =

6 √ η, π 2

T˜ =

kT , ε

λ=

1 + 0.2977T˜ 1 + 0.3316T˜ + 0.0010477T˜ 2

(10)

where ε is the interaction energy between segments; T˜ the reduced temperature; k the Boltzmann constant. 2.2. Equation of state based on SLP model The second equation of state for chainlike fluids can be obtained by using Song et al.’s treatment [4] 4r  a  Zpert = − η (11) b kT where


 2π 3 kT a= σ εFa , 3 ε


 2π 3 kT b= σ Fb 3 ε

(12)

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with

 
3/2  kT kT Fa = 1.8681 exp −0.0619 + 0.6715 exp −1.7317 ε ε  

1/2 
3/2  kT kT kT = 0.7307 exp −0.1649 Fb + 0.2697 exp −2.3973 ε ε ε In the SLP model, the dimensionless diameter λ is set to 3 Fb (kT/ε).


kT ε








(13)

(14)

2.3. Equation of state based on O’LC model By applying O’Lenick and Chiew model [28] to Eq. (1), we obtain the third equation of state. Compressibility factor of perturbation term can be written 
  12rη 1 pert Z = JA(η,r) + (15) − 1 JB(η,r) λ6 T˜ λ6 where (−1.0755 − 0.22169r) + (2.077 − 4.5236r)η − (1.6623 − 4.41r)η2 0.4571 + r

 0.4213 0.1974 JB(η,r) = 0.03171 + + 1.3253 + η + 5.3598η2 r r
1/6 3 2 0.005397r + 0.006354 0.01222r + 0.005102 −1 = T˜ + λ 9.44 + r 9.947 + r JA(η,r) =

(16) (17)

(18)

Obviously, the effective hard-sphere diameter in O’LC model depends on both temperature and chain length. 2.4. Equation of state based on GWL model Based on the SW coordination number model [29,30], perturbation term is 
 N0 Ω − 1 Ωα ρ∗ pert Z = −r + 2α α T˜ √ where α is the constant, set to −1/ 2, and N0 , α, ρ∗ , Ω are defined as 4π ∗3 1 − ρ∗ (λ − 1)ρ∗ , α= √ , 3 2 

 1 − ρ∗ ε¯ ε , ε¯ = Ω = exp √ kT 2

N0 =

ρ∗ = rρσ 3 =

6η π

(19)

(20) (21)

where λ∗ is a parameter characterizing the well width and is set to 1.5. Obviously, λ = 1 in GWL model.

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2.5. Equation of state based on CK model We rewrite the form of Chen and Kreglewski model [16] to compare conveniently with other models. The perturbation term of equation of state based on CK model is Z

pert

=r

9 4



 nDmn

m=1 n=1

η n  u m 0.74048 kT

(22)

where Dmn is the universal constant; u/k the temperature-dependent energy parameter and defined by  e  (23) u=ε 1+ T k where e/k is the constant and depend on substance type. For polymer, e/k = 10. The dimensionless hard-sphere diameter in CK model can be determined by
 −3ε λ = 1 − 0.12 exp kT

(24)

2.6. Equation of state based on GS model By introducing GS model [10] based on second-order perturbation theory into Eq. (1), we obtained GS type equation of state Zpert = Z1 + Z2

(25)

where

∂(ηI1 ) 2  ε  3 r σ ∂η kT     ∂(ηI2 ) ε 2 3 Z2 = −πρrC1 σ − C2 ηI2 (r, η) r2 ∂η kT Z1 = −2πρ

∂(ηI1 )

= aj (r)(j + 1)ηj , ∂η j=0 6

6

I1 (r, η) = aj (r)ηj ,

(26) (27)

∂(ηI2 )

bj (r)(j + 1)ηj = ∂η j=0

I2 (r, η) =

6

6

bj (r)ηj

(28)

(29)

j=0

j=0

aj (r) = a0j +

r−1 r−1r−2 a1j + a2j r r r

(30)

bj (r) = b0j +

r−1 r−1r−2 b1j + b2j r r r

(31)




8η − 2η2 20η − 27η2 + 12η3 − 2η4 C1 = 1 + r + (1 − r) (1 − η)4 [(1 − η)(2 − η)]2

−1 (32)

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C2 = C1

−4η2 + 20η + 8 2η3 + 12η2 − 48η + 40 r + (1 − r) (1 − η)5 [(1 − η)(2 − η)]3

 (33)

a0j , a1j , a2j , b0j , b1j , b2j are the constant. Obviously, the dimensionless diameter of GK model is equal to 1. Similar to other model [1–12], above six equations of state contain three molecular parameters, namely, segment number (chain length) r, segment diameter (temperature-independent) σ, and interaction energy between segments ε/k. These parameters are all obtained by fitting pVT data of pure substance.

3. Simplified equation of state—molar mass independent equation For polymer with sufficiently high molar mass, pVT data are usually insensitive to the chain length. Besides, the molar mass (M) of polymers in some references is not given. We have noted that the chain length r is increased as the molar mass increasing. In this case, although r and M are assumed to be large, the ratio of chain length and molar mass r/M remains finite. we apply parameter r/M to replace r and use the equation of state in limit r → ∞ to obtain the model parameters. Based on such idea, equation of state can be simplified as P˜ a η + b η2 − c η3 = + Zpert (1 − η)3 ηT˜

(34)

and p˜ =

pπ(NAv σ 3 ) 3 λ, 6R(ε/k)

a = 0.70951,

η=

π (NAv σ 3 )(r/M) 3 λ 6 v

b = 6.60081,

c = 7.5871

(35) (36)

NAv and R are the Avogadro’s number and the gas constant, respectively; v is the specific volume. 3.1. Simplified equation I—S-CSP type equation of state From Eqs. (7) and (34), the simplified CSP type equation of state (S-CSP) is   P˜ a η + b η2 − c η3 Z1 Z2 = + + (1 − η)3 ηT˜ T˜ T˜ 2

(37)

3.2. Simplified equation II—S-SLP type equation of state By simplifying to SLP type equation of state, the final form (S-SLP) is given P˜ a η + b η2 − c η3 4η Fa (ε/kT) = − (1 − η)3 ηT˜ T˜ Fb (ε/kT)

(38)

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133

3.3. Simplified equation III—S-O’LC type equation of state Following the same treatment as discussed above, O’LC type equation of state can be modified as 
  P˜ a η + b η2 − c η3 12η 1 = λA(η,r) + + (39) − 1 λB(η,r) (1 − η)3 λ6 ηT˜ T˜ λ6 Here λA(η,r) = −0.22169 − 4.5236η + 4.41η2

(40)

λB(η,r) = 0.03171 + 1.3253η + 5.3598η2
1/6 3 2 − 1 = 0.005397T˜ + 0.01222 λ

(41) (42)

This modified equation is called S-O’LC type equation of state. 3.4. Simplified equation IV—S-GWL type equation of state S-GWL type equation of state can be expressed as  
P˜ a η + b η2 − c η3 N0 Ω − 1 Ωα ρ∗ = − + (1 − η)3 2α α ηT˜ T˜

(43)

3.5. Simplified equation V—S-CK type equation of state It is very easy to obtain S-CK type equation of state 9 4  η n  u m P˜ a η + b η2 − c η3

= + nD mn (1 − η)3 0.74048 kT ηT˜ m

(44)

n

3.6. Simplified equation VI—S-GS type equation of state After simple treatment to Gross and Sadowski model [10], we obtained a S-GS type equation of state     P˜ a η + b η2 − c η3 12η ∂(ηI1 ) 6η ∂(ηI2 ) = − − C − (C r) ηI (r, η) (45) 1 2 2 (1 − η)3 ∂η ∂η ηT˜ T˜ T˜ 2 where




C1 r = C2 =

8η − 2η2 20η − 27η2 + 12η3 − 2η4 − (1 − η)4 [(1 − η)(2 − η)]2

−1

(−4η2 + 20η + 8/(1 − η)5 ) − (2η3 + 12η2 − 48η + 40/[(1 − η)(2 − η)]3 ) (8η − 2η2 /(1 − η)4 ) − (20η − 27η2 − 12η3 − 2η4 /[(1 − η)(2 − η)]2 )

aj (r) = a0j + a1j + a2j

(46) (47) (48)

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bj (r) = b0j + b1j + b2j

(49)

The equation of state based on Alder–Young–Mark model [15] has been detailed given in the literature [33]. Its simplified version is called S-AYM in this work. Obviously, the right side of the simplified equation of state does not contain parameter r. The modified equations of state possess three parameters: r/M, NAv σ 3 and ε/k. They can be obtained by fitting pVT data of pure polymer. Note that the simplified equations of state are applicable only to polymers with high molecular weight (molar mass), at low molecular weight, the assumption of r → ∞ no longer holds. Strongpoint of the simplified equation is that it is applicable to polymers with unknown molecular weight. The pVT data, except PA series and poly(ether sulphone) [35], to estimate the model parameters are regenerated by the Tait equation with parameters from Rodgers [36]. Relative deviations between

Fig. 1. pVT of poly(ether sulphone). (䉫) Experimental data [35]; lines: calculated results by equations of state. (A) Solid lines: S-AYM; short-dash lines: S-CK; long-dash lines: S-O’LC; dot-dash lines: S-CSP. (B) Solid lines: S-GS; short-dash lines: S-AYM; long-dash lines: S-SLP; dot-dash lines: S-GWL. p/(kg/cm2 ): 1—0; 2—400; 3—800; 4—1200; 5—1600; 6—2000.

C. Peng et al. / Fluid Phase Equilibria 206 (2003) 127–145

calculated and experimental values for specific volumes of polymers is given by  Nm  calc  v (Ti , pi ) − vexp (Ti , pi )  1

  × 100 ∆(%) =  Nm i=1  vexp (Ti , pi )

135

(50)

Fig. 1 shows pVT of poly(ether sulphone). Diamonds are experimental data [35]. Lines are the calculated results by different equations of state. The results of S-AYM are all illustrated in part A and part B of figure to show comparison between S-AYM and other equations of state. The relative deviation of the specific volumes is 0.13% for S-CSP, 0.14% for S-SLP, 0.14% for S-AYM, 0.23% for S-O’LC, 0.21% for S-GWL, 0.15% for S-CK and 0.12% for S-GS, respectively. pVT of Nylon 6 is

Fig. 2. pVT of Nylon 6. (䉫) Experimental data [35]; lines: calculated results. (A) Solid lines: S-CK; short-dash lines: S-O’LC; long-dash lines: S-CSP; dot-dash lines: S-AYM. (B) Solid lines: S-GS; short-dash lines: S-SLP; long-dash lines: S-GWL; dot-dash lines: S-AYM. p/MPa: 1—0; 2—40; 3—80; 4—120; 5—160; 6—200.

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illustrated in Fig. 2. The lowest deviation of the specific volumes is obtained by S-GS, followed by S-CSP, S-CK, S-AYM, S-O’LC, S-GWL and S-SLP. It was found that the best correlation is given by S-GS. Tables 1–6 give the model parameters and correlated results of the specific volumes for 50 polymers by using different equations of state. The results from S-AYM have been shown in literature [33]. It is shown that excellent results for the calculation of the specific volumes of polymers can be obtained, while any perturbation term discussed above is used. The accuracy of correlation depends on the choice of perturbation terms, but the S-GS and S-CSP give best results. The total mean deviation is 0.148% for S-GS, 0.168% for S-CSP, 0.207% for S-AYM, 0.216% for S-CK, 0.221% for S-SLP, 0.261% for S-O’LC and 0.301% for S-GWL, respectively. It is worth noticing that the parameter r/M increases as parameter σ 3 decreases polymers. This implies that the product of r/M and σ 3 obtained by any equation of state may be same for a given polymer. This case is explicit illustrated in Fig. 3. In the figures, the vertical coordinate represents the product of r/M and σ 3 , the number on the horizontal coordinate represents substance serial number coupled with Table 1. We note that all equations of state except S-SLP has almost same value of (r/M) × σ 3 for a given polymer. This phenomenon shows that equations of state discussed above have the same description for molecular volume, because parameter (r/M) × σ 3 inflects the volume per chain molecule rather than per segment. On the other hand, (r/M)×σ 3 of S-SLP has the same change tendency as that of others. Another parameter, the product of r/M and ε/k, gives the energy per chain molecule as shown in Fig. 4. Evidently, value of energy has a significant difference between equations of state based on different perturbation theory. On the average, S-GS shows the smallest interaction energy, and the biggest one is obtained by S-SLP. From a practical perspective, the S-GWL and S-CSP is of much simpler formulation and is even easier to use. However, advances in statistical mechanics and an increase of computer power have allowed the development of equations of state that are somewhat complex formulation. Synthetically, the S-GS and S-CSP equations can be recommend for pure polymer.

Fig. 3. The product of r/M and σ 3 for 50 polymers. (䉫) S-CSP; ( ) S-SLP; ( ) S-O’LC; (×) S-GWL; (䊊) S-CK; (+) S-GS; (䊐) S-AYM.

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Table 1 Model parameters and correlated results of pVT data for 50 polymers by using S-CSP Serial number

Polymer

Symbol

ε/k (K)

NAv σ 3 × 106 (m3 /mol)

r/M

∆ (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 35 37 38 39 40 41 42 43 44 45 46 47 48 49 50

a-Polypropylene Bisphenol chloral polycarbonate Branched polyethylene High-density polyethylene Hexafluoro bisphenol-A polycarbonate High MW linear polyethylene i-Poly(1-butene) i-Polypropylene Low-density polyethylene Low-density polyethylene “A” Low-density polyethylene “B” Low-density polyethylene “C” Linear polyethylene Polyarylate (Ardel) Poly(acrylic acid) cis-1,4-Polybutadiene Bisphenol-A polycarbonate Poly(cyclohexyl methacrylate) Poly(ε-caprolactone) Poly(dimethylsiloxane) Poly(ethyl acrylate) Poly(epichlorohydrin) Poly(ether ether ketone) Poly(ethyl methacrylate) Poly(ethylene oxide) Phenoxy Poly(methyl acrylate) Poly(methyl methacrylate) Poly(4-methyl-1-pentene) Poly(o-methylstyrene) Poly(2,6-dimethylphenylene oxide) Polystyrene Polysulfone Poly(tetrahydrofuran) Poly(vinyl acetate) Poly(vinyl chloride) Poly(vinyl methyl ether) Tetramethyl bisphenol-A polycarbonate Polyethersulphone Polyamide 11 Polyamide 12 Polyamide 6 Polyamide 6/10 Polyamide 6/12 Polyamide 6/6 Polyamide 6/7 Polyamide 6/8 Polyamide 6/9 Polyamide 7 Polyamide 9

a-PP BCPC BPE HDPE HFPC HMLPE i-PB i-PP LDPE LDPE-A LDPE-B LDPE-C LPE PAr PAA PBD PC PcHMA PCL PDMS PEA PECH PEEK PEMA PEO PH PMA PMMA PMP PoMS PPO PS PSF PTHF PVAc PVC PVME TMPC PESP PA11 PA12 PA6 PA6/10 PA6/12 PA6/6 PA6/7 PA6/8 PA6/9 PA7 PA9

244.055 307.921 256.633 296.056 244.470 223.539 278.479 282.087 287.206 270.323 279.529 278.380 242.471 277.737 395.988 271.343 299.902 277.890 289.401 205.379 257.810 295.324 285.621 251.720 263.220 305.976 273.296 326.132 240.983 345.635 258.542 354.286 325.686 281.425 256.692 328.249 277.664 271.790 358.298 288.516 281.402 393.765 275.389 288.241 272.860 306.444 297.949 302.856 311.112 283.772

21.8737 18.1241 20.0462 28.4611 15.5315 13.8298 25.7731 28.5937 24.754 21.2255 22.3430 22.4163 17.3933 12.3638 16.6566 19.8276 17.0137 16.4509 22.2763 21.6349 18.6139 19.333 13.5943 14.2054 16.4274 15.6530 17.1025 20.3542 25.7419 25.9668 16.0059 28.7342 17.0944 22.6385 15.9378 22.9127 19.1642 17.8891 18.1810 17.1351 17.2732 31.9946 17.8785 17.1386 10.9819 15.4782 15.7084 16.3485 16.9555 15.6766

0.0449677 0.033397 0.0410374 0.0367343 0.0342320 0.0552978 0.0393754 0.0361901 0.0418772 0.0478212 0.0458567 0.0456424 0.0456424 0.0563420 0.0379348 0.048811 0.0416187 0.0467383 0.0363022 0.0387736 0.0411663 0.0332930 0.047156 0.0528551 0.0468843 0.047739 0.0432649 0.0364142 0.0386551 0.033241 0.046098 0.0297731 0.0402097 0.0394984 0.045213 0.0278189 0.0443533 0.0416187 0.0345735 0.0508999 0.0510920 0.0222267 0.0489631 0.0495638 0.069147 0.0521003 0.0519323 0.0508999 0.0486590 0.0537810

0.18 0.22 0.16 0.17 0.24 0.16 0.21 0.24 0.15 0.18 0.17 0.16 0.16 0.13 0.07 0.05 0.19 0.14 0.09 0.17 0.20 0.10 0.17 0.11 0.09 0.23 0.16 0.06 0.72 0.09 0.18 0.08 0.14 0.10 0.06 0.11 0.16 0.17 0.13 0.21 0.22 0.15 0.33 0.24 0.16 0.18 0.20 0.16 0.17 0.26

N m

Note: ∆(%) = (1/Nm ) [35], the other [36].

exp i=1 |(vi

exp

− vcalc i )/vi | × 100, Nm is number of data. Data sources: PA series and poly(ether sulphone)

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Table 2 Model parameters and correlated results of pVT data for 50 polymers by using S-SLP Serial number

Polymer

ε/k (K)

NAv σ 3 × 106 (m3 /mol)

r/M

∆ (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 35 37 38 39 40 41 42 43 44 45 46 47 48 49 50

a-PP BCPC BPE HDPE HFPC HMLPE i-PB i-PP LDPE LDPE-A LDPE-B LDPE-C LPE Par PAA PBD PC PcHMA PCL PDMS PEA PECH PEEK PEMA PEO PH PMA PMMA PMP PoMS PPO PS PSF PTHF PVAc PVC PVME TMPC PESP PA11 PA12 PA6 PA6/10 PA6/12 PA6/6 PA6/7 PA6/8 PA6/9 PA7 PA9

328.478 420.987 359.303 400.031 356.953 326.554 386.076 400.950 380.753 371.989 380.261 379.284 348.533 402.845 464.071 316.422 419.149 377.664 290.900 279.414 343.635 387.011 421.788 204.036 350.020 401.725 357.912 210.812 356.709 430.886 386.097 429.765 453.321 358.104 324.849 398.920 355.754 308.716 495.897 415.821 407.671 358.449 396.857 421.219 340.491 441.412 432.854 436.768 443.932 418.674

31.3211 27.1071 31.6198 41.1175 27.7228 24.4245 39.7538 46.9152 34.2126 32.1111 32.8649 33.1474 29.4352 21.8032 17.7996 20.8613 26.9276 24.4316 15.7855 31.7183 26.1257 26.0507 25.5116 5.42302 22.9102 21.2333 23.0164 4.66035 48.7974 31.3605 30.3751 32.8017 26.7838 28.4434 19.8952 26.1778 24.5556 16.4051 28.0901 29.4333 29.8354 14.8236 30.6913 30.5087 13.0428 26.5977 27.5722 28.0909 28.2432 28.6987

0.0484304 0.0345735 0.0405902 0.0392523 0.0303479 0.0489631 0.0396955 0.0345735 0.0465302 0.0490396 0.0482028 0.0477466 0.0424486 0.0504217 0.0529428 0.069147 0.0409724 0.048659 0.0721860 0.0409070 0.0450285 0.0378148 0.0399436 0.171386 0.0515709 0.053781 0.0491133 0.186953 0.0324607 0.0416187 0.0387736 0.0391908 0.0399436 0.047739 0.0549533 0.0366083 0.0526003 0.0646252 0.0347896 0.0466742 0.0466742 0.0778664 0.0448975 0.0440812 0.0860844 0.0477466 0.0467383 0.0466742 0.0458567 0.0466742

0.20 0.23 0.18 0.18 0.28 0.18 0.23 0.28 0.16 0.21 0.20 0.18 0.19 0.13 0.07 0.07 0.21 0.14 0.38 0.20 0.28 0.09 0.20 0.68 0.19 0.25 0.24 0.77 0.81 0.10 0.21 0.09 0.17 0.13 0.08 0.11 0.22 0.45 0.14 0.23 0.26 0.43 0.37 0.26 0.28 0.20 0.22 0.18 0.18 0.28

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139

Table 3 Model parameters and correlated results of pVT data for 50 polymers by using S-O’LC Serial number

Polymer

ε/k (K)

NAv σ 3 × 106 (m3 /mol)

r/M

∆ (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 35 37 38 39 40 41 42 43 44 45 46 47 48 49 50

a-PP BCPC BPE HDPE HFPC HMLPE i-PB i-PP LDPE LDPE-A LDPE-B LDPE-C LPE PAr PAA PBD PC PcHMA PCL PDMS PEA PECH PEEK PEMA PEO PH PMA PMMA PMP PoMS PPO PS PSF PTHF PVAc PVC PVME TMPC PESP PA11 PA12 PA6 PA6/10 PA6/12 PA6/6 PA6/7 PA6/8 PA6/9 PA7 PA9

336.870 494.083 401.192 495.583 359.502 330.643 439.969 432.133 474.217 428.996 450.297 447.834 305.983 412.755 753.546 516.861 467.669 448.150 493.254 235.486 421.518 557.961 406.086 301.868 433.393 511.393 457.673 580.425 337.746 610.769 379.104 645.158 503.747 469.391 441.074 598.546 480.850 404.176 518.046 434.216 420.566 447.808 414.650 427.806 396.654 462.027 443.927 457.599 476.071 420.117

17.2855 20.0476 21.2353 33.0417 15.3458 13.7804 27.6003 29.3971 27.7680 22.9646 24.5011 24.6090 11.6484 12.5812 22.0716 25.8745 18.1559 18.1671 25.9144 11.2074 20.9971 26.9764 12.9244 8.42328 18.8958 18.4118 19.9111 24.9117 23.5039 31.4450 15.7886 35.7511 18.0063 25.2572 18.6864 28.7746 23.2117 17.9541 16.3622 17.5082 17.6599 13.8852 18.2207 17.1268 10.7468 15.8739 15.8377 16.7789 17.6983 15.7262

0.056868 0.0315933 0.0402097 0.0335011 0.0352138 0.0565181 0.038295 0.0363022 0.0393754 0.0460980 0.0438091 0.0435370 0.0654205 0.0565181 0.0312134 0.0407835 0.0404638 0.0443465 0.033189 0.0700113 0.038415 0.0260112 0.0497196 0.0845130 0.0429928 0.0429928 0.0393754 0.0319737 0.0421833 0.0294515 0.0474425 0.0258501 0.0394984 0.0374546 0.0411016 0.0239343 0.0391293 0.0423159 0.0388952 0.0510599 0.0510920 0.0579849 0.0492672 0.0505798 0.0714839 0.0521003 0.0526003 0.0508999 0.0480508 0.0546234

0.31 0.32 0.23 0.25 0.34 0.21 0.33 0.37 0.22 0.29 0.29 0.27 0.31 0.22 0.11 0.09 0.34 0.21 0.14 0.42 0.39 0.12 0.21 0.31 0.24 0.41 0.33 0.11 0.90 0.15 0.26 0.14 0.26 0.21 0.13 0.16 0.32 0.23 0.23 0.30 0.32 0.21 0.49 0.30 0.19 0.25 0.26 0.24 0.21 0.35

140

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Table 4 Model parameters and correlated results of pVT data for 50 polymers by using S-GWL Serial number

Polymer

ε/k (K)

NAv σ 3 × 106 (m3 /mol)

r/M

∆ (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 35 37 38 39 40 41 42 43 44 45 46 47 48 49 50

a-PP BCPC BPE HDPE HFPC HMLPE i-PB i-PP LDPE LDPE-A LDPE-B LDPE-C LPE PAr PAA PBD PC PcHMA PCL PDMS PEA PECH PEEK PEMA PEO PH PMA PMMA PMP PoMS PPO PS PSF PTHF PVAc PVC PVME TMPC PESP PA11 PA12 PA6 PA6/10 PA6/12 PA6/6 PA6/7 PA6/8 PA6/9 PA7 PA9

297.684 517.648 440.268 494.959 430.204 394.009 474.471 488.448 471.383 457.674 468.041 466.632 417.416 487.365 572.270 390.841 512.876 465.742 402.898 344.062 423.284 484.039 402.088 428.149 430.630 494.329 440.377 501.055 428.271 531.376 456.206 531.890 555.094 443.040 403.509 496.574 437.971 479.348 607.924 504.785 494.404 520.142 481.556 508.171 477.611 535.886 523.384 529.605 541.372 500.059

10.7572 17.7854 20.8402 26.4987 18.7598 16.6779 25.9930 30.8277 21.9070 21.1074 21.3018 21.4092 19.0399 14.8827 10.5882 12.2490 17.9111 15.7614 14.2234 20.3964 16.9178 16.3485 10.1566 14.4786 14.9007 13.7233 14.6245 14.9885 34.0848 19.1299 19.6085 19.7452 17.8795 17.9625 12.5029 16.0343 15.3269 21.3045 18.4784 19.7191 20.2337 15.6473 20.6746 20.4811 13.4958 17.8721 18.5277 18.7533 18.9128 18.9818

0.0758878 0.0316123 0.0369024 0.0370185 0.0267224 0.0432649 0.0364142 0.0314703 0.0436731 0.0447649 0.0446254 0.0443533 0.0390167 0.0440812 0.0534449 0.0707476 0.0369024 0.0452835 0.0500956 0.0381749 0.0417479 0.0363022 0.0534449 0.0483549 0.0475908 0.0499395 0.0463861 0.0448975 0.0276428 0.0409708 0.0355179 0.0391293 0.035854 0.0454281 0.0526003 0.035966 0.0505798 0.0329809 0.0316924 0.0416187 0.0411016 0.0472304 0.0398195 0.0391353 0.0529428 0.0424486 0.0414894 0.0417479 0.0409708 0.0419044

0.53 0.37 0.27 0.29 0.40 0.23 0.38 0.43 0.25 0.35 0.35 0.33 0.26 0.25 0.13 0.12 0.39 0.24 0.28 0.29 0.51 0.15 0.34 0.18 0.36 0.49 0.45 0.12 0.98 0.18 0.29 0.19 0.30 0.27 0.15 0.18 0.26 0.26 0.21 0.34 0.37 0.23 0.56 0.34 0.21 0.28 0.29 0.27 0.23 0.40

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Table 5 Model parameters and correlated results of pVT data for 50 polymers by using S-CK Serial number

Polymer

ε/k (K)

NAv σ 3 × 106 (m3 /mol)

r/M

∆ (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 35 37 38 39 40 41 42 43 44 45 46 47 48 49 50

a-PP BCPC BPE HDPE HFPC HMLPE i-PB i-PP LDPE LDPE-A LDPE-B LDPE-C LPE PAr PAA PBD PC PcHMA PCL PDMS PEA PECH PEEK PEMA PEO PH PMA PMMA PMP PoMS PPO PS PSF PTHF PVAc PVC PVME TMPC PESP PA11 PA12 PA6 PA6/10 PA6/12 PA6/6 PA6/7 PA6/8 PA6/9 PA7 PA9

310.231 397.781 272.579 407.559 322.729 250.716 352.882 342.415 391.774 348.498 367.428 364.832 291.548 324.731 569.021 387.253 375.031 367.350 405.147 269.301 344.926 443.134 317.875 322.880 354.013 418.527 374.146 441.863 265.509 492.211 297.143 507.970 403.477 384.468 368.329 469.721 386.984 317.440 458.122 326.424 330.860 354.468 326.938 310.520 293.817 365.114 349.496 359.300 379.596 313.828

23.5842 20.2172 14.3684 35.5435 19.2703 11.4301 27.8755 28.3042 30.2328 23.7514 25.8260 25.8459 16.8899 11.3230 20.2400 23.4002 17.8538 19.2153 27.8666 25.2094 22.2690 27.1155 10.9572 15.8218 19.8603 19.2172 20.9827 23.2105 20.4888 32.1856 14.0228 34.9071 17.4821 26.9352 20.7955 28.1852 23.9041 16.1954 19.8075 14.3037 15.8688 12.8399 16.9676 12.5766 12.3241 14.6014 14.2463 15.2411 16.9599 12.3338

0.0412955 0.0297265 0.0534449 0.0297265 0.0277319 0.0635254 0.0359660 0.0356299 0.0345735 0.0424486 0.0396155 0.0394984 0.0457139 0.0592231 0.0315124 0.0421833 0.0390152 0.0400156 0.0294515 0.0332930 0.0345735 0.0244816 0.0552117 0.0470824 0.0390152 0.0392523 0.0356299 0.0321447 0.0457139 0.0272957 0.0503397 0.0249697 0.0385350 0.0335011 0.0354058 0.0230490 0.0360781 0.044219 0.0313723 0.0579849 0.0536129 0.0592231 0.0499395 0.0630246 0.0508999 0.0534449 0.0551255 0.0527642 0.0474425 0.0644252

0.22 0.28 0.26 0.22 0.21 0.16 0.25 0.26 0.20 0.23 0.23 0.21 0.16 0.16 0.11 0.13 0.26 0.17 0.14 0.20 0.33 0.14 0.17 0.11 0.19 0.37 0.29 0.12 0.76 0.16 0.19 0.18 0.20 0.21 0.12 0.18 0.32 0.17 0.15 0.26 0.24 0.18 0.37 0.29 0.24 0.19 0.21 0.18 0.17 0.33

142

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Table 6 Model parameters and correlated results of pVT data for 50 polymers by using S-GS Serial number

Polymer

ε/k (K)

NAv σ 3 × 106 (m3 /mol)

r/M

∆ (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 35 37 38 39 40 41 42 43 44 45 46 47 48 49 50

a-PP BCPC BPE HDPE HFPC HMLPE i-PB i-PP LDPE LDPE-A LDPE-B LDPE-C LPE PAr PAA PBD PC PcHMA PCL PDMS PEA PECH PEEK PEMA PEO PH PMA PMMA PMP PoMS PPO PS PSF PTHF PVAc PVC PVME TMPC PESP PA11 PA12 PA6 PA6/10 PA6/12 PA6/6 PA6/7 PA6/8 PA6/9 PA7 PA9

335.302 420.559 352.014 407.967 347.910 316.571 382.124 391.785 389.266 370.394 381.260 379.980 337.848 393.610 552.926 379.637 413.710 378.528 359.871 279.858 352.149 428.938 373.289 344.669 359.325 417.191 372.826 444.109 360.228 471.804 368.549 487.893 445.892 376.437 349.095 451.784 378.697 385.052 490.311 403.632 396.247 414.257 387.169 407.421 392.463 428.219 418.864 422.986 430.240 402.580

43.1469 34.9306 39.6161 56.4167 33.6122 29.5561 51.3180 59.3860 46.8729 41.8282 43.2153 43.4998 35.8671 25.8124 30.3401 36.2863 33.3345 31.2327 34.3397 42.2017 35.5952 40.7160 20.8412 27.5973 31.3015 29.2522 32.3032 36.9324 65.1631 47.8572 34.7869 52.9292 33.0754 40.8159 29.3512 42.0644 35.7954 38.4291 35.0956 35.3432 36.2385 27.4715 37.2172 36.3733 23.7379 31.6344 32.5620 33.0358 33.8832 33.4506

0.0236252 0.017927 0.0215124 0.0192203 0.0165945 0.0271008 0.0204880 0.0181227 0.0228329 0.0251298 0.0245196 0.0243295 0.023049 0.028259 0.0216469 0.0277308 0.0220406 0.0254499 0.0237633 0.0205520 0.0222687 0.0165945 0.0309241 0.0281710 0.0254499 0.0264143 0.023893 0.0207441 0.0163360 0.0186541 0.0222687 0.0167502 0.0215123 0.0225430 0.0253699 0.0157009 0.0245576 0.0202951 0.0185411 0.0257300 0.0254499 0.0297716 0.0245576 0.0242233 0.0337092 0.0265704 0.0261682 0.0262503 0.0252899 0.0264143

0.16 0.23 0.11 0.15 0.17 0.07 0.18 0.17 0.15 0.17 0.18 0.15 0.08 0.06 0.10 0.06 0.19 0.14 0.11 0.11 0.26 0.08 0.17 0.08 0.12 0.31 0.23 0.07 0.52 0.10 0.08 0.10 0.14 0.13 0.07 0.13 0.25 0.08 0.12 0.16 0.17 0.14 0.27 0.16 0.11 0.14 0.15 0.13 0.14 0.20

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Fig. 4. The product of r/M and ε/k for 50 polymers. (䉫) S-CSP; ( ) S-SLP; ( ) S-O’LC; (×) S-GWL; (䊊) S-CK; (+) S-GS; (䊐) S-AYM.

4. Conclusions We reviewed several well-known perturbation terms based on different theory. Six versions of equations of state were proposed based on different perturbation terms. The chain formation term of the established equations adopts Hu et al.’s model. We simplified these equations of state to apply conveniently to polymers. Further, we obtained model parameters in each equations of state by fitting pVT data for 50 polymers. Like other equations of state, the simplified equations of state were found to estimate polymer specific volumes accurately. From our observations, the S-GS and S-CSP equations are recommend for pure polymer. Further results on fluid mixtures and polymer solutions will be reported in the next paper. List of symbols a, b, c parameters or variable d temperature-dependent segment diameter k Boltzmann constant M molecular weight p˜ reduced pressure r segment length T absolute temperature (K) T˜ reduced temperature u temperature-dependent energy parameter v specific volume y(2e) effective cavity correlation function Z compressibility factor Greek letters ε temperature-independent energy parameter ε¯ effective energy parameter η packing density

144

λ λ∗ ρ σ

C. Peng et al. / Fluid Phase Equilibria 206 (2003) 127–145

dimensionless hard-sphere diameter parameter characterizing the well width number density of chain molecules temperature-independent segment diameter

Superscripts calc calculated results exp experimental results pert perturbation term Acknowledgements This work was supported by the National Natural Science Foundation of China (Project No. 20025618, 29976011), the Doctoral Research Foundation sponsored by the Ministry of Education of China (Project No. 1999025103) and the Education Committee of Shanghai City of China. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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