Pressure-volume-temperature properties for binary oligomeric blends of polyethylene glycol mono-laurate with polyethylene glycol mono-4-octylphenyl ether or polyethylene glycol mono-4-nonylphenyl ether at pressures up to 50 MPa

J. Chem. Thermodynamics 135 (2019) 215–224

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Pressure-volume-temperature properties for binary oligomeric blends of polyethylene glycol mono-laurate with polyethylene glycol mono-4octylphenyl ether or polyethylene glycol mono-4-nonylphenyl ether at pressures up to 50 MPa Chien-Fong Lu, Ho-mu Lin, Ming-Jer Lee ⇑ Department of Chemical Engineering, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei 106-07, Taiwan

a r t i c l e

i n f o

Article history: Received 6 March 2019 Received in revised form 27 March 2019 Accepted 29 March 2019 Available online 30 March 2019 Keywords: Density PEG ethers PEG ester Equations of state

a b s t r a c t Densities are measured for two binary systems of oligomeric blends, polyethylene glycol mono-4octylphenyl ether) (PEGOPE) + polyethylene glycol mono-laurate (PEGML) and polyethylene glycol mono-4-nonylphenyl ether (PEGNPE) + PEGML, at temperatures from 298.15 K to 348.15 K and pressures up to 50 MPa. The excess volumes are negative for these two binary systems over the majority of composition range, except for the regions near pure components, in which the values may change into positive. Tait equation represents accurately for the pressure-effect on the liquid densities. Additionally, a generalized equation with two characteristic parameters correlates well the P-V-T data over the entire experimental conditions for each binary system. Those P-V-T data are also correlated satisfactorily with the Flory-Orwoll-Vrij (FOV) and the Schotte equations of state with one binary interaction parameter for each oligomeric blend system. Ó 2019 Elsevier Ltd.

1. Introduction Polyethylene glycol esters and ethers are widely used as nonionic surfactants in industries. These two classes of surfactants have a hydrophilic oxide chain and a hydrocarbon hydrophobic group. It may involve all kinds of intermolecular interactions, including electrostatic, induction, dispersion, and hydrogen bonding, in the mixtures containing these polymeric compounds. Pressure-volume-temperature (PVT) properties of such mixtures provide valuable experimental evidences for exploring the interactions in the mixtures, and also can be applied to determine the model parameters of equations of state (EOS) and then use these models to estimate various thermodynamic properties of the mixtures for industrial applications. Moreover, the PVT data are formed a fundamental basis to develop advanced thermodynamic models for polymeric systems. Using a high-pressure vibrating U-tube densimeter, our research group conducted a series of PVT data measurements for various polymeric systems at different temperatures and up to

⇑ Corresponding author. E-mail address: [email protected] (M.-J. Lee). https://doi.org/10.1016/j.jct.2019.03.038 0021-9614/Ó 2019 Elsevier Ltd.

50 MPa [1–9]. Most of the investigated mixtures contain the fractionation cuts of oligomeric polyethylene glycols (PEG), polypropylene glycols (PPG), and polyethylene glycol methyl ethers (PEGME). Recently the studies on the PVT properties of the related polymeric systems at atmospheric pressure have also been reported [10–15]. However, studies on the volumetric behavior of the oligomeric blends of PEG esters and PEG ethers over a wide range of pressure are still limited. In the present study, the oligomer of polyethylene glycol monolaurate (PEGML) is selected as a model compound of PEG esters, and those of polyethylene glycol mono-4-octylphenyl ether) (PEGOPE) and polyethylene glycol mono-4-nonylphenyl ether (PEGNPE) are taken as model compounds of PEG ethers. The densities are measured for two oligomeric blends, PEGOPE + PEGML and PEGNPE + PEGML, at temperatures from 298.15 K to 348.15 K and pressures up to 50 MPa. No literature data are available at the comparable conditions, except for the densities of PEGOPE [9]. The excess volumes of each binary system are calculated from the PVT data which reveal the polar and hydrogen-bonding effects on the volumetric behavior of those oligomeric blends. These new obtained PVT data are also correlated with two EOS, the FOV [16] and the Schotte [17], and the performances of these two EOS are compared.

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C.-F. Lu et al. / J. Chem. Thermodynamics 135 (2019) 215–224

study. Without the correction of viscosity effect, the uncertainty of the density measurements was estimated to be 4.0  104 g
cm3.

2. Experimental Polyethylene glycol mono-laurate (PEGML, CAS: 9004-81-3, Mn = 525 g
mol1, Mw/Mn = 1.043), poly(ethylene glycol mono-4octylphenyl ether) (PEGOPE, CAS: 9002-93-1, Mn = 576 g
mol1, Mw/Mn = 1.015), and polyethylene glycol mono-4-nonylphenyl ether (PEGNPE, CAS: 26027-38-3, Mn = 460 g
mol1, Mw/ Mn = 1.021) were purchased from Tokyo Chemical Industry Co., Ltd. (TCI) (Japan). Their number-average molar mass (Mn) and the polydispersity (Mw/Mn) were measured by using a MatrixAssisted Laser Desorption/Ionization Time of Flight (MALDI-TOF). Each oligomeric compound has been degassed before use. The description of the materials is given in Table 1. The schematic diagram of the PVT measurement apparatus has been shown in Lee et al. [4]. Each liquid mixture sample was prepared from the degassed oligomeric compounds. By using an electronic balance, the uncertainty of the composition of each sample is estimated to be about 0.0004 in mass fraction (wi). A high-pressure vibrating U-tube densitometer (DMA 512 P, Anton Paar, Austria) was used for the density (q) measurements. While a high pressure generator (Model-62-6-10, High Pressure Equipment Company, USA) was employed to manually adjust the pressure in the measuring cell, a pressure transducer (Model-PDCR 911, 0–70 MPa, Druck, UK) with a digital indicator (model-DPI 261, Druck, UK) displayed the measured pressure (P). The uncertainty of pressure measurement is estimated to be 0.1% as pressures higher than 0.1 MPa and 3% at atmospheric pressure (0.1 MPa). A thermostatic circulator maintains the temperature of the measuring cell (T) to within ±0.05 K and the temperature was measured by a precision digital thermometer (Model-1560, Hart Scientific, USA) with a platinum RTD probe to an uncertainty of 0.02 K. The oscillation period (ti) of sample i was read by DMA 48 densimeter (Anton Paar, Austria) and the value can be converted into density (qi) by











qi = g
cm3 ¼ A t2i  B

ð1Þ

where A and B are apparatus parameters. The values of A and B were determined by using two calibration fluids: pure water [18] and dry nitrogen [19]. The calibration runs were conducted at each measuring temperature over pressure range of 0.1 MPa to 50 MPa. The calibration can be reproduced water densities to an average absolute relative deviation (AARD) of 0.01% over the whole calibration conditions. The viscosity differences between the samples and the calibration fluids might deteriorate the accuracy of density measurement by an oscillating densitometer (Ashcroft et al. [20]), but the effect is generally minor under the conditions of the present

3. Results and discussion In the present study, the PVT properties of PEGOPE + PEGML and PEGNPE + PEGML mixtures and their constituent oligomers were determined experimentally at 298.15 K, 318.15 K, and 348.15 K over pressures from 0.1 MPa to 50 MPa. Since only the density data of PEGOPE are available in literature [9], the comparison of experimental results with the literature data is made in Table 2. The agreement is better than ±0.1%. While Table 3 lists the densities of PEGOPE, PEGNPE, and PEGML. Tables 4 and 5 present the PVT properties and the excess volumes for PEGOPE + PEGML and PEGNPE + PEGML, respectively. The Tait equation was adopted to represent the densities varying with pressure at given temperature and composition. Its expression is defined as

ðq  qo Þ=q ¼ C ln½ðD þ PÞ=ðD þ 0:1Þ

ð2Þ

where qo represents the density at 0.1 MPa. The values of parameters C and D were determined by fitting the density data to the Tait equation by minimization of the objective function p1:

"

p1 ¼

# n   X   qk;calc  qk;expt =qk;expt =n

ð3Þ

k ¼1

where n equals the number of data points, and qk,calc and qk,expt are the calculated and the experimental densities of the kth point, respectively. The calculated results are reported in Supplementary Information Tables S1 and S2 for PEGOPE + PEGML and PEGNPE + PEGML, respectively. Fig. 1 shows the isothermal densities of PEGNPE (1) + PEGML (2), x1 = 0.5330, varying with pressure at temperatures from 318.15 K to 348.15 K. The smoothed lines represent the calculated results from the Tait equation, indicating that the agreement between correlated and experimental values is within experimental uncertainty. The isothermal compressibility jT at given temperature, pressure and composition can be readily calculated from its definition embedded with the Tait equation:





jT = MPa1 ¼ ð1=V Þð@V=@PÞT;x ¼ ðV o =V Þ½C=ðD þ PÞ

ð4Þ

where Vo is the corresponding specific volume at 0.1 MPa. The values of C and D have been reported in Tables S1 and S2. Fig. 2 is an illustrative example for the mixture of PEGNPE (1) + PEGML (2) with x1 = 0.5530. As shown from the graph, the isothermal com-

Table 1 Description of the materials.a Chemical name (CAS number)

Molecular structure

PEGOPE (9002-93-1) C8H17C6H4

O

PEGNPE (26027-38-3) C9H19C6H4

O

H

H

C

C

H

H

H

H

C

C

H

PEGML (9004-81-3) CH3(CH2)10CO

a

O

Mn/(g
mo11)

Polydispersity (MW/Mn)

Supplier

Purification method

576

1.015

TCI, Japan

degas

460

1.021

TCI, Japan

degas

525

1.043

TCI, Japan

degas

OH

n

n

OH

H H

H

C

C

H

H

n

OH

The variables Mn and Mw are number average and weight average molar mass, respectively.

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C.-F. Lu et al. / J. Chem. Thermodynamics 135 (2019) 215–224 Table 2 Comparison of experimental density data from this work with literature values.a Substance

T/K

P/MPa

qexpt/(g
cm3)

qlit/(g
cm3) [9]

PEGOPE

298.15

0.1 20 40 50 0.1 20 40 50 0.1 20 40 50

1.0640 1.0736 1.0824 1.0865 1.0478 1.0581 1.0676 1.0721 1.0243 1.0359 1.0463 1.0513

1.0636 1.0731 1.0818 1.0860 1.0475 1.0580 1.0673 1.0717 1.0248 1.0362 1.0468 1.0519

318.15

348.15

a Standard uncertainties of temperature T, and density q are u(T) = 0.05 K; u(q) = 4.0  104 g
cm3. Relative standard uncertainties of pressure P are ur(P) = 0.001 and 0.03 for pressures higher than 0.1 MPa and at atmospheric pressure, respectively.

Table 3 Experimental results of density measurement for pure components.a

P/MPa

T/K = 298.15 q/(g
cm3)

T/K = 318.15 q/(g
cm3)

T/K = 348.15 q/(g
cm3)

PEGML 0.1 10 15 20 25 30 35 40 45 50

1.0368 1.0420 1.0445 1.0470 1.0495 1.0518 1.0540 1.0563 1.0585 1.0607

1.0224 1.0280 1.0308 1.0335 1.0362 1.0387 1.0412 1.0437 1.0459 1.0483

0.9980 1.0044 1.0074 1.0104 1.0134 1.0162 1.0189 1.0216 1.0242 1.0268

PEGOPE 0.1 10 15 20 25 30 35 40 45 50

1.0640 1.0689 1.0712 1.0736 1.0757 1.0780 1.0803 1.0824 1.0844 1.0865

1.0478 1.0530 1.0556 1.0581 1.0606 1.0629 1.0652 1.0676 1.0699 1.0721

1.0243 1.0302 1.0331 1.0359 1.0387 1.0413 1.0439 1.0463 1.0489 1.0513

PEGNPE 0.1 10 15 20 25 30 35 40 45 50

0.9956 1.0008 1.0031 1.0056 1.0079 1.0102 1.0125 1.0146 1.0168 1.0189

0.9802 0.9857 0.9883 0.9910 0.9935 0.9959 0.9985 1.0009 1.0031 1.0054

0.9581 0.9643 0.9674 0.9704 0.9733 0.9760 0.9787 0.9813 0.9838 0.9863





jTo = MPa1 ¼ C=ðD þ 0:1Þ

ð6Þ

where the Tait constants C and D have been given in Tables S1 and S2. This empirical model, Eq. (5), correlates density (q) data to a root mean square deviation (RMSD) of 1.4  104 g.cm3 for PEGOPE + PEGML (with d1 = 1.5209 and d2 = 1.0812) and 1.7  104 g.cm3 for PEGNPE + PEGML (with d1 = 1.5819 and d2 = 1.0903). Fig. 3 shows all the PVT data of each binary system can be generalized by Eq. (5). This empirical model, Eq. (5), could be applied to estimate the densities at elevated pressures with known specific volume and isothermal compressibility at atmospheric pressure (Vo and jTo) as the characteristic parameters (d1 and d2) have been determined from a few high-pressure PVT data points at any given compositions. The tabulated excess molar volumes (VE) in Tables 4 and 5 were calculated from the experimental PVT data via: 1

V E =ðcm3 mol Þ ¼ V m  ðx1 V 01 þ x2 V 02 Þ

ð7Þ

where xi and Vio are the mole fraction and molar volume for component i, respectively. The variable Vm is the molar volume of a mixture which was calculated from the following equation: 1

V m =ðcm3 mol Þ ¼ ðx1 M1 þ x2 M 2 Þ=q

a Standard uncertainties of temperature T, and density q are u(T) = 0.05 K; u(q) = 4.0  104 g
cm3. Relative standard uncertainties of pressure P are ur(P) = 0.001 and 0.03 for pressures higher than 0.1 MPa and at atmospheric pressure, respectively.

pressibility decreases with increasing pressure and increases with increasing temperature. Lin et al. [5] modified a generalized model of Sanchez et al. [21] for representing the PVT properties of solvent, polymers, and their mixtures over wide range of conditions:

ðP  0:1ÞjTo ¼ d1 ½ðV o =V Þ  1d2

mental range by using two characteristic parameters, d1 and d2, for each binary system. The value of jTo at given temperature and composition can be calculated from Eq. (6), which is derived from Eq. (4) by introducing V = Vo and P = 0.1 MPa.

ð5Þ

where jTo and Vo are the isothermal compressibility and the specific volume at 0.1 MPa, respectively. The PVT data can be merged onto a single curve of (P  0.1) jTo against (Vo/V)  1 over entire experi-

ð8Þ

where Mi is the number-average molar mass of component i. The uncertainty of the calculated excess volumes was estimated to be 0.08 cm3
mol1. Figs. 4 and 5 depict the variation of the isothermal excess volumes with composition for PEGOPE + PEGML and PEGNPE + PEGML, respectively. The part (a) of these two graphs is at P = 0.1 MPa and temperatures from 298.15 K to 348.15 K and part (b) at T = 298.15 K and pressures from 0.1 MPa to 40 MPa. The excess volumes are negative for these two binary systems over a majority of composition range, except for the regions near pure components, where the values may change into positive. It is suggested that the specific (chemical) interactions (hydrogenbonding) may be dominant over the majority of composition range, resulting in volume contraction. At fixed composition and pressure, the magnitudes of volume contraction decrease with an increase of temperature. It is consistent with the hydrogen bonding effect becoming weaker as temperature increase. The excess volumes were correlated with a Redlich-Kister type equation:

V E =ðx1 x2 Þ ¼

j X k¼0

Ak ðx1  x2 Þk

ð9Þ

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C.-F. Lu et al. / J. Chem. Thermodynamics 135 (2019) 215–224

Table 4 Experimental results of density measurement and excess volumes for the oligomeric blends PEGOPE (1) + PEGML (2).a x1 = 0.0921b

T/K = 298.15 3

T/K = 318.15

P/MPa

q/(g
cm )

V /(cm
mol

0.1 10 15 20 25 30 35 40 45 50

1.0397 1.0449 1.0473 1.0497 1.0521 1.0544 1.0568 1.0590 1.0612 1.0633

x1 = 0.1857 0.1 10 15 20 25 30 35 40 45 50 x1 = 0.2810 0.1 10 15 20 25 30 35 40 45 50

1

3

T/K = 348.15

q/(g
cm3)

V E /(cm3
mol1)

0.01 0.02 0.03 0.05 0.07 0.08 0.10 0.12 0.14 0.16

1.0007 1.0070 1.0099 1.0129 1.0157 1.0185 1.0212 1.0238 1.0264 1.0289

0.11 0.14 0.16 0.18 0.19 0.21 0.22 0.24 0.26 0.27

1.0278 1.0332 1.0359 1.0385 1.0411 1.0435 1.0459 1.0483 1.0506 1.0528

0.23 0.20 0.19 0.19 0.18 0.16 0.13 0.09 0.07 0.05

1.0036 1.0099 1.0128 1.0158 1.0186 1.0214 1.0241 1.0267 1.0293 1.0318

0.08 0.07 0.06 0.04 0.04 0.03 0.02 0.01 0.01 0.02

1.0306 1.0360 1.0387 1.0413 1.0438 1.0463 1.0486 1.0510 1.0533 1.0555

0.39 0.37 0.35 0.33 0.32 0.31 0.28 0.25 0.23 0.22

1.0064 1.0126 1.0156 1.0185 1.0214 1.0241 1.0268 1.0293 1.0319 1.0344

0.21 0.20 0.20 0.18 0.16 0.16 0.15 0.13 0.11 0.09

q/(g
cm )

V /(cm
mol

0.16 0.15 0.13 0.13 0.11 0.10 0.10 0.08 0.07 0.05

1.0248 1.0303 1.0330 1.0356 1.0382 1.0406 1.0430 1.0455 1.0478 1.0500

1.0429 1.0480 1.0504 1.0528 1.0551 1.0575 1.0598 1.0620 1.0641 1.0662

0.40 0.36 0.34 0.34 0.34 0.33 0.31 0.29 0.25 0.22

1.0460 1.0511 1.0534 1.0558 1.0581 1.0605 1.0628 1.0649 1.0671 1.0692

0.60 0.57 0.53 0.53 0.51 0.51 0.49 0.45 0.45 0.44

E

3

)

E

3

1

)

x1 = 0.3780

T/K = 298.15

P/MPa

q/(g
cm3)

V E /(cm3
mol1)

q/(g
cm3)

V E /(cm3
mol1)

q/(g
cm3)

V E /(cm3
mol1)

0.1 10 15 20 25 30 35 40 45 50

1.0488 1.0538 1.0561 1.0585 1.0608 1.0631 1.0654 1.0676 1.0697 1.0719

0.67 0.58 0.57 0.57 0.54 0.52 0.50 0.50 0.48 0.48

1.0333 1.0387 1.0414 1.0439 1.0464 1.0488 1.0512 1.0536 1.0559 1.0581

0.47 0.46 0.45 0.41 0.39 0.38 0.36 0.34 0.32 0.30

1.0091 1.0153 1.0182 1.0211 1.0240 1.0266 1.0293 1.0318 1.0344 1.0369

0.29 0.27 0.25 0.24 0.23 0.21 0.18 0.17 0.16 0.14

x1=0.4772 0.1 10 15 20 25 30 35 40 45 50

1.0518 1.0567 1.0591 1.0615 1.0637 1.0660 1.0684 1.0706 1.0727 1.0748

0.77 0.73 0.72 0.72 0.68 0.66 0.67 0.67 0.66 0.66

1.0361 1.0414 1.0441 1.0467 1.0492 1.0517 1.0540 1.0564 1.0587 1.0609

0.61 0.59 0.59 0.58 0.56 0.56 0.55 0.53 0.52 0.50

1.0118 1.0180 1.0209 1.0238 1.0266 1.0293 1.0320 1.0345 1.0371 1.0396

0.36 0.34 0.32 0.32 0.31 0.30 0.28 0.28 0.27 0.26

x1=0.5777 0.1 10 15 20 25 30 35 40 45 50

1.0542 1.0592 1.0615 1.0639 1.0662 1.0685 1.0708 1.0729 1.0750 1.0771

0.64 0.61 0.59 0.58 0.58 0.57 0.56 0.54 0.52 0.52

1.0383 1.0436 1.0462 1.0487 1.0513 1.0536 1.0560 1.0583 1.0606 1.0628

0.41 0.40 0.38 0.34 0.36 0.33 0.30 0.27 0.26 0.25

1.0143 1.0203 1.0232 1.0261 1.0289 1.0315 1.0343 1.0367 1.0393 1.0417

0.27 0.24 0.24 0.21 0.20 0.16 0.18 0.16 0.15 0.14

T/K = 318.15

T/K = 348.15

219

C.-F. Lu et al. / J. Chem. Thermodynamics 135 (2019) 215–224 x1 = 0.6803

T/K = 298.15

P/MPa

q/(g
cm3)

T/K = 318.15 V /(cm
mol

0.1 10 15 20 25 30 35 40 45 50

1.0565 1.0613 1.0636 1.0660 1.0683 1.0705 1.0729 1.0750 1.0770 1.0791

x1=0.7848 0.1 10 15 20 25 30 35 40 45 50 x1=0.8913 0.1 10 15 20 25 30 35 40 45 50

T/K = 348.15

q/(g
cm3)

V /(cm
mol

0.42 0.34 0.32 0.32 0.30 0.28 0.27 0.25 0.23 0.21

1.0406 1.0459 1.0485 1.0510 1.0535 1.0559 1.0582 1.0605 1.0628 1.0651

1.0589 1.0638 1.0661 1.0685 1.0707 1.0730 1.0753 1.0774 1.0794 1.0815

0.24 0.20 0.19 0.19 0.19 0.17 0.16 0.15 0.15 0.12

1.0612 1.0660 1.0683 1.0707 1.0729 1.0751 1.0774 1.0795 1.0815 1.0836

0.03 0.04 0.05 0.07 0.09 0.09 0.09 0.12 0.14 0.16

E

3

1

)

q/(g
cm3)

V E /(cm3
mol1)

0.29 0.27 0.27 0.23 0.23 0.22 0.19 0.17 0.16 0.15

1.0167 1.0227 1.0256 1.0285 1.0313 1.0339 1.0366 1.0390 1.0416 1.0441

0.16 0.14 0.14 0.13 0.12 0.09 0.10 0.09 0.08 0.07

1.0429 1.0482 1.0508 1.0532 1.0557 1.0581 1.0604 1.0628 1.0651 1.0673

0.14 0.14 0.12 0.10 0.10 0.09 0.08 0.06 0.06 0.05

1.0191 1.0251 1.0279 1.0308 1.0335 1.0362 1.0388 1.0412 1.0438 1.0462

0.06 0.03 0.02 0.00 0.02 0.03 0.05 0.04 0.07 0.10

1.0450 1.0503 1.0529 1.0553 1.0577 1.0601 1.0624 1.0647 1.0670 1.0692

0.09 0.10 0.11 0.15 0.16 0.16 0.19 0.20 0.20 0.20

1.0213 1.0272 1.0301 1.0329 1.0357 1.0383 1.0409 1.0433 1.0459 1.0483

0.19 0.22 0.22 0.23 0.24 0.24 0.25 0.26 0.29 0.30

E

3

1

)

a Standard uncertainties of temperature T, mole fraction of component 1 x1, density q, and molar excess volume V E are u(T) = 0.05 K; u(P) = 0.02 MPa; u(x1) = 0.0004; u(q) = 4.0  104 g
cm3; u(V E ) = 0.08 cm3
mol1. Relative standard uncertainties of pressure P are ur(P) = 0.001 and 0.03 for pressures higher than 0.1 MPa and at atmospheric pressure, respectively. b x1: mole fraction of component 1; calculated with the average molar mass of 576 g.mol1 and 525 g.mol1 for PEGOPE and PEGML, respectively.

Table 5 Experimental results of density measurement and excess volumes for the oligomeric blends PEGNPE (1) + PEGML (2).a x1 = 0.1127b

T/K = 298.15

P/MPa

q/(g
cm3)

V E /(cm3
mol1)

q/(g.cm3)

V E /(cm3
mol1)

q/(g
cm3)

V E /(cm3
mol1)

0.1 10 15 20 25 30 35 40 45 50

1.0330 1.0382 1.0406 1.0430 1.0454 1.0477 1.0501 1.0523 1.0545 1.0567

0.25 0.25 0.23 0.23 0.21 0.19 0.20 0.18 0.16 0.16

1.0181 1.0235 1.0263 1.0289 1.0315 1.0339 1.0364 1.0388 1.0411 1.0434

0.05 0.05 0.04 0.02 0.00 0.03 0.03 0.05 0.06 0.09

0.9941 1.0004 1.0035 1.0065 1.0095 1.0123 1.0150 1.0176 1.0203 1.0229

0.06 0.06 0.03 0.01 0.01 0.00 0.00 0.01 0.02 0.02

x1 = 0.2221 0.1 10 15 20 25 30 35 40 45 50

1.0294 1.0346 1.0370 1.0394 1.0418 1.0441 1.0465 1.0486 1.0509 1.0530

0.56 0.54 0.53 0.52 0.51 0.50 0.48 0.47 0.45 0.43

1.0143 1.0198 1.0225 1.0251 1.0277 1.0301 1.0326 1.0350 1.0374 1.0396

0.33 0.32 0.31 0.29 0.28 0.25 0.25 0.23 0.22 0.20

0.9902 0.9966 0.9996 1.0026 1.0055 1.0082 1.0109 1.0135 1.0161 1.0186

0.10 0.09 0.09 0.09 0.07 0.05 0.03 0.02 0.01 0.01

x1 = 0.3286 0.1 10 15 20 25 30

1.0255 1.0306 1.0330 1.0354 1.0378 1.0402

0.67 0.66 0.65 0.63 0.63 0.63

1.0101 1.0156 1.0184 1.0210 1.0236 1.0260

0.41 0.41 0.40 0.38 0.37 0.35

0.9862 0.9926 0.9956 0.9985 1.0014 1.0041

0.15 0.15 0.14 0.11 0.10 0.09

T/K = 318.15

T/K = 348.15

(continued on next page)

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C.-F. Lu et al. / J. Chem. Thermodynamics 135 (2019) 215–224

Table 5 (continued) x1 = 0.1127b

T/K = 298.15

P/MPa

q/(g
cm3)

V E /(cm3
mol1)

q/(g.cm3)

V E /(cm3
mol1)

q/(g
cm3)

V E /(cm3
mol1)

35 40 45 50

1.0425 1.0447 1.0469 1.0491

0.61 0.60 0.59 0.59

1.0285 1.0309 1.0332 1.0355

0.34 0.33 0.33 0.33

1.0069 1.0094 1.0120 1.0146

0.07 0.06 0.06 0.05

T/K = 318.15

x1 = 0.4322

T/K = 298.15

P/MPa

q/(g
cm3)

V /(cm
mol

0.1 10 15 20 25 30 35 40 45 50

1.0216 1.0267 1.0291 1.0315 1.0339 1.0362 1.0385 1.0407 1.0429 1.0451

x1 = 0.5330 0.1 10 15 20 25 30 35 40 45 50 x1 = 0.6313 0.1 10 15 20 25 30 35 40 45 50 x1 = 0.7271

T/K = 348.15

T/K = 318.15

T/K = 348.15

q/(g
cm3)

V /(cm
mol

0.79 0.78 0.77 0.75 0.73 0.71 0.70 0.70 0.69 0.69

1.0060 1.0115 1.0142 1.0168 1.0194 1.0218 1.0243 1.0267 1.0290 1.0313

1.0178 1.0229 1.0253 1.0277 1.0301 1.0324 1.0347 1.0368 1.0391 1.0412

0.96 0.92 0.92 0.90 0.89 0.87 0.87 0.84 0.84 0.84

1.0139 1.0191 1.0215 1.0239 1.0262 1.0285 1.0308 1.0329 1.0351 1.0373 T/K = 298.15

1.06 1.03 1.03 1.01 1.00 1.00 0.97 0.95 0.93 0.93

E

3

1

)

q / (g
cm3)

V E /cm3
mol1)

0.47 0.47 0.45 0.43 0.42 0.41 0.40 0.39 0.37 0.37

0.9824 0.9887 0.9917 0.9947 0.9976 1.0003 1.0031 1.0056 1.0082 1.0107

0.28 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.18 0.18

1.0022 1.0077 1.0104 1.0130 1.0156 1.0180 1.0205 1.0229 1.0251 1.0274

0.66 0.65 0.64 0.62 0.62 0.61 0.59 0.57 0.55 0.55

0.9787 0.9850 0.9880 0.9910 0.9939 0.9967 0.9994 1.0019 1.0045 1.0070

0.47 0.45 0.43 0.43 0.41 0.41 0.40 0.37 0.36 0.33

0.9984 1.0039 1.0066 1.0093 1.0118 1.0142 1.0167 1.0191 1.0214 1.0236 T/K = 318.15

0.88 0.87 0.86 0.85 0.83 0.81 0.79 0.77 0.76 0.76

0.9752 0.9814 0.9845 0.9875 0.9903 0.9931 0.9958 0.9983 1.0009 1.0034 T/K = 348.15

0.69 0.67 0.65 0.65 0.63 0.63 0.62 0.60 0.59 0.57

E

3

1

)

P/MPa

q/(g
cm3)

V E /(cm3
mol1)

q/(g
cm3)

V E /(cm3
mol)

q/(g
cm3)

V E /(cm3
mol1)

0.1 10 15 20 25 30 35 40 45 50

1.0095 1.0146 1.0169 1.0193 1.0217 1.0240 1.0263 1.0284 1.0306 1.0328

0.86 0.83 0.82 0.80 0.80 0.79 0.78 0.76 0.75 0.74

0.9938 0.9993 1.0020 1.0046 1.0072 1.0096 1.0121 1.0145 1.0167 1.0190

0.66 0.65 0.65 0.64 0.62 0.61 0.59 0.58 0.56 0.55

0.9708 0.9771 0.9801 0.9831 0.9860 0.9887 0.9914 0.9939 0.9965 0.9990

0.52 0.49 0.47 0.47 0.46 0.44 0.43 0.41 0.39 0.38

x1 = 0.8205 0.1 10 15 20 25 30 35 40 45 50

1.0051 1.0102 1.0125 1.0150 1.0172 1.0195 1.0218 1.0239 1.0261 1.0282

0.69 0.67 0.66 0.64 0.62 0.60 0.59 0.58 0.57 0.55

0.9894 0.9949 0.9975 1.0002 1.0027 1.0051 1.0076 1.0100 1.0122 1.0145

0.55 0.51 0.50 0.49 0.48 0.47 0.45 0.44 0.43 0.40

0.9667 0.9730 0.9760 0.9790 0.9818 0.9845 0.9873 0.9898 0.9923 0.9948

0.44 0.42 0.39 0.37 0.36 0.35 0.34 0.33 0.32 0.31

x1 = 0.9115 0.1 10 15 20 25 30

1.0009 1.0060 1.0083 1.0108 1.0131 1.0154

0.59 0.58 0.57 0.57 0.56 0.55

0.9853 0.9908 0.9934 0.9961 0.9986 1.0010

0.50 0.50 0.49 0.49 0.48 0.47

0.9627 0.9689 0.9720 0.9750 0.9778 0.9805

0.36 0.33 0.33 0.32 0.32 0.31

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C.-F. Lu et al. / J. Chem. Thermodynamics 135 (2019) 215–224 35 40 45 50

1.0177 1.0198 1.0220 1.0241

0.55 0.54 0.53 0.53

0.47 0.46 0.45 0.43

1.0035 1.0059 1.0082 1.0104

0.9833 0.9858 0.9883 0.9908

0.31 0.30 0.29 0.29

a Standard uncertainties of temperature T, mole fraction of component 1 x1, density q, and molar excess volume VE are u(T) = 0.05 K; u(P) = 0.02 MPa; u(x1) = 0.0004; u(q) = 4.0  104 g
cm3; u(VE) = 0.08 cm3
mol1. Relative standard uncertainties of pressure P are ur(P) = 0.001 and 0.03 for pressures higher than 0.1 MPa and at atmospheric pressure, respectively. b x1: mole fraction of component 1; calculated with the average molar mass of 460 g
mol1 and 525 g
mol1 for PEGNPE and PEGML, respectively.

The optimal values of coefficients Ak were determined by minimization of the objective function p2:

p2

" # n  X  E E   ¼ V k;calc  V k;expt  =n

ð10Þ

k¼1

Fig. 1. Isothermal densities varying with pressure for PEGNPE (1) + PEGML (2) with x1 = 0.5330 (, 298.15 K; r, 318.15 K; ▲, 348.15 K; —, calculated values from the Tait equation).

Fig. 2. Isothermal compressibility varying with pressure for PEGNPE + PEGML with x1 = 0.5330 (- - - -, 298.15 K; – . –, 318.15 K; —, 348.15 K).

Fig. 3. Correlated results from Eq. (5), (a) PEGOPE + PEGML; (b) PEGNPE + PEGML at temperatures from 298.15 K to 348.15 K and pressure from 0.1 MPa to 50 MPa over entire composition range (e, expt.; —, calculated values from Eq. (5)).

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C.-F. Lu et al. / J. Chem. Thermodynamics 135 (2019) 215–224

where n represents the number of data points. The correlated results (Ao to A2, and RMSD VE) are listed in Supplementary Information Tables S3 and S4 for PEGOPE + PEGML and PEGNPE + PEGML, respectively, and illustrated as the dashed curves in Figs. 4 and 5. E

3

1

3

and 0.121 cm The maximum RMSD of V are 0.090 cm mol mol1 for PEGOPE + PEGML and PEGNPE + PEGML, respectively.

 



 1=3

P V =T ¼ V

  1=3 1    1 V 1  T V

ð11Þ

Schotte EOS:

    1=3 1     1=3 þ V 1 P V = T ¼ ðRT  =P MV  Þ 1  V    1  T V

4. Correlation of specific volumes with equations of state

ð12Þ 



where M is the number-average molar mass, P = P/P*, V = V/V* and

The experimental specific volumes of the binary mixtures and their constituent components were correlated with the FloryOrwoll-Vrij (FOV) [16] and the Schotte [17] EOS, which are defined as FOV EOS:

T = T/T*. The parameters P*, V*, and T* are characteristic pressure, specific volume, and temperature, respectively. The optimal values of these characteristic parameters for each oligomeric component

Fig. 4. The excess volumes varying with composition for PEGOPE (1) + PEGML (2) at (a) P = 0.1 MPa and T at (, 298.15 K; r, 318.15 K; ▲, 348.15 K); - - - -, calculated values from the Redlich-Kister equation. (b) T = 298.15 K and P at (, 0.1 MPa; r, 20 MPa; ▲, 40 MPa); - - - -, calculated values from the Redlich-Kister equation.

Fig. 5. The excess volumes varying with composition for PEGNPE (1) + PEGML (2) at (a) P = 0.1 MPa and T at (, 298.15 K; r, 318.15 K; ▲, 348.15 K); - - - -, calculated values from the Redlich-Kister equation. (b) T = 298.15 K and P at (, 0.1 MPa; r, 20 MPa; ▲, 40 MPa); - - - -, calculated values from the Redlich-Kister equation.



223

C.-F. Lu et al. / J. Chem. Thermodynamics 135 (2019) 215–224 Table 6 Characteristic parameters of the FOV and the Schotte EOS for the oligomeric components.a Substance

V*/(cm3
g1)

P*/(MPa)

T*/(K)

104 RMSD Vb/(cm3
g1)

FOV EOS PEGOPE PEGNPE PEGML

0.7876 0.8413 0.8080

666.28 595.32 606.10

6222.0 6204.9 6225.7

3.9 4.6 5.5

Schotte EOS PEGOPE PEGNPE PEGML

0.7794 0.8322 0.7973

660.92 614.50 625.94

5575.6 5562.9 5520.1

3.5 3.9 5.3

a

The variables V*, P*, and T* are characteristic volume, pressure, and temperature, respectively.
 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 RMSD V= cm3
g 1 ¼ k¼1 ðV k;calc  V k;expt Þ =ðn  nparam Þ where n and nparam are the number of data points and the number of parameters, respectively.

b

Table 7 Correlated results from FOV and Schotte EOS for the oligomeric blends. Mixture (1) + (2)

EOS

D12a

rb/%

PEGOPE + PEGML

FOV Schotte

0.0063 0.0050

0.04 0.05

PEGNPE + PEGML

FOV Schotte

0.0116 0.0113

0.05 0.05

a

The variable D12 is the binary interaction parameter. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2
ffi Pn
where n and r=% ¼ 100 %  k¼1 V k;calc  V k;expt =V k;expt = n  nparam nparam are the number of data points and the number of parameters, respectively. b

were determined by fitting the experimental PVT data to the EOS with the objective function p3:

p3

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n uX ¼t ðV k;calc  V k;expt Þ2 =ðn  nparam Þ

ð13Þ

k¼1

where V, n, and nparam are specific volume, the number of data points, and the number of parameters, respectively. Table 6 reports the calculated results, indicating that the FOV and the Schotte equations correlated well the specific volumes of the pure compounds. The root mean square deviations (RMSD) are no greater than 0.00055 cm3
g1. The mixing rules of Schotte [17] were taken to calculate the properties of oligomeric blends, which are defined as

" V m ¼ M m

c X




Wi = Mi V i



!#1 ð14Þ

i¼1

T m ¼ Pm =

c X






Wi Pi =T i



ð15Þ

i¼1

and

Pm ¼

c X c X i¼1

Wi Wj Pij

ð16Þ

j¼1

with

Wi ¼ wi V i =

c X

! wk V k

ð17Þ

k¼1

and

0:5
 Pij ¼ 1  Dij Pi Pj

ð18Þ

where Mi, Wi, and wi denote the number-average molar mass, the segment volume fraction, and the weight fraction of component i, respectively. The parameter Dij in Eq. (18) is a binary interaction

Fig. 6. Comparison of the calculated values from the equations of state with experimental data for (a) PEGOPE (1) + PEGML(2) with x1 = 0.5777; (b) PEGNPE (1) + PEGML (2) with x1 = 0.5330 at (, 298.15 K; r, 318.15 K; ▲, 348.15 K); —, calculated values from the FOV EOS; - - - -, calculated values from the Schotte EOS.

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C.-F. Lu et al. / J. Chem. Thermodynamics 135 (2019) 215–224

parameter for i-j pair. The optimal value of Dij for each binary oligomeric blend was determined by correlating the experimental PVT data to the EOS by using the objective function p4:

p4

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n uX ¼t ðV k;calc  V k;expt Þ2 =ðn  nparam Þ

ð19Þ

k¼1

Table 7 shows that comparable results were obtained by using these two EOS. The average deviations are no greater than 0.05% for both two binary systems. Fig. 6(a) and (b) compare the calculated densities with the experimental values for PEGOPE (1) + PEGML (2) with x1 = 0.5777 and PEGNPE (1) + PEGML (2) with x1 = 0.5330, respectively. The calculated values, either from the FOV or from the Schotte, agree satisfactorily with the experimental results for these two oligomeric blends. 5. Conclusions The PVT properties have been measured for two oligomeric blend systems of PEGOPE + PEGML and PEGNPE + PEGML at temperatures from 298.15 K to 348.15 K and pressures up to 50 MPa. Over the whole experimental pressure range, the Tait equation represents accurately for each liquid density isotherm. By using two characteristic parameters, an empirical generalized equation can satisfactorily correlate the PVT data over the entire experimental conditions for each binary system. The excess volumes were found to be negative over the majority of composition range, revealing that volume contraction occurs after the blending of the oligomers. The FOV and the Schotte EOS are capable of correlating the specific volumes of these two oligomeric blend systems to average deviations no greater than 0.05%. Acknowledgment The financial support from the Ministry of Science and Technology (MOST), Taiwan, through grant MOST-105-2221-E011-144-MY3 is gratefully acknowledged. Notes The authors declare no competing financial interest. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jct.2019.03.038. References [1] M.J. Lee, C.K. Lo, H.M. Lin, PVT measurements for mixtures of poly(ethylene glycol methyl ether) with poly(ethylene glycol) from 298 K to 338 K and pressures up to 30 MPa, J. Chem. Eng. Data 43 (1998) 1076–1081.

[2] J.S. Chang, M.J. Lee, H.M. Lin, PVT of fractionation cuts of poly(ethylene glycol) and poly(propylene glycol) from 298 K to 338 K and pressures up to 30 MPa, J. Chem. Eng. Japan 32 (1999) 611–618. [3] M.J. Lee, C.K. Lo, H.M. Lin, PVT measurements for mixtures of 1-octanol with oligomeric poly(ethylene glycol) from 298 K to 338 K and pressures up to 30 MPa, J. Chem. Eng. Data 44 (1999) 1379–1385. [4] M.J. Lee, Y.C. Tuan, H.M. Lin, Pressure, volume, and temperature for mixtures of poly(ethylene glycol methyl ether)-350 + anisole and poly(ethylene glycol)200 + anisole from 298 K to 338 K and pressures up to 50 MPa, J. Chem. Eng. Data 45 (2000) 1100–1104. [5] H.M. Lin, T.S. Hsu, M.J. Lee, Pressure-volume-temperature properties for binary polymer solutions of poly(propylene glycol) with 1-octanol and acetophenone, Macromolecules 34 (2001) 6297–6303. [6] M.J. Lee, Y.C. Tuan, H.M. Lin, Pressure-volume-temperature properties for binary and ternary polymer solutions of poly(ethylene glycol), poly(propylene glycol), and poly(ethylene glycol methyl ether) with anisole, Polymer 44 (2003) 3891–3900. [7] M.J. Lee, T.S. Hsu, Y.C. Tuan, H.M. Lin, Pressure-volume-temperature properties for 1-octanol + acetophenone, poly(propylene glycol) + 1-octanol + acetophenone, and poly(ethylene glycol) + poly(rthylene glycol methyl ether) + anisole, J. Chem. Eng. Data 49 (2004) 1052–1058. [8] M.J. Lee, K.L. Ho, H.M. Lin, Pressure-volume-temperature properties for binary oligomeric solutions of poly(ethylene glycol) and poly(ethylene glycol methyl ether) with acetophenone up to 50 MPa, J. Chem. Eng. Data 51 (2006) 1115– 1121. [9] M.J. Lee, T.J. Ku, H.M. Lin, Pressure-volume-temperature properties for binary oligomeric solutions of poly(ethylene glycol mono-4-octylphenyl ether) with 1-octanol or acetophenone at pressures up to 50 MPa, J. Chem. Thermodyn. 41 (2009) 1178–1185. [10] Z.P. Visak, L.M. Ilharco, A.R. Garcia, V. Najdanovic-Visak, J.M.N.A. Fareleira, F.J. P. Caetano, M.L. Kijevcanin, Slobodan P. Serbanovic, Volumetric properties and spectroscopic studies of pyridine or nicotine solutions in liquid polyethylene glycols, J. Phys. Chem. B 115 (2011) 8481–8492. [11] R. Sadeghi, H.B. Kahaki, Thermodynamics of aqueous solutions of poly ethylene glycol di-methyl ethers in the presence or absence of ammonium phosphate salts, Fluid Phase Equilib. 306 (2011) 219–228. [12] M.T. Zafarani-Moattar, N. Tohidifar, Effect of temperature on volumetric and transport properties of ternary polyethylene glycol di-methyl ether + poly ethylene glycol 400 + water and the corresponding binary aqueous solutions: measurement and correlation, Fluid Phase Equilib. 343 (2013) (2000) 43–57. [13] M.T. Zafarani-Moattar, S. Dehghanian, Intermolecular interactions in mixtures of poly (ethylene glycol) with methoxybenzene and ethoxybenzene: volumetric and viscometric studies, J. Chem. Thermodyn. 71 (2014) 221–230. [14] M.T. Zafarani-Moattar, N. Tohidifar, Study of thermodynamic and transport properties of aqueous system containing poly(ethylene glycol) dimethyl ether 2000 and poly(propylene glycol) 400, J. Mol. Liquids 207 (2015) 80–89. [15] S. Ebrahimi, R. Sadeghi, Density, speed of sound, and viscosity of some binary and ternary aqueous polymer solutions at different temperatures, J. Chem. Eng. Data 60 (2015) 3132–3147. [16] P.J. Flory, R.A. Orwoll, A. Vrij, Statistical thermodynamics of chain molecule liquids. I. An equation of state for normal paraffin hydrocarbons, J. Am. Chem. Soc. 86 (1964) 3507–3514. [17] W. Schotte, Vapor-liquid equilibrium calculations for polymer solutions, Ind. Eng. Chem. Process Des. Dev. 21 (1982) 289–296. [18] L. Haar, J.S. Gallagher, G.S. Kell, NBS/NRC Steam Tables: Thermodynamic and Transport Properties and Computer Programs for Vapor and Liquid States of Water in SI Units, Hemisphere, New York, 1984. [19] N.B. Vargaftik, Tables on the Thermodynamical Properties of Liquids and Gases, 2nd ed., Hemisphere, Washington, DC, 1975. [20] S.J. Ashcroft, D.R. Booker, J.C.R. Turner, Density measurement by oscillating tube. Effects of viscosity, temperature, calibration and signal processing, J. Chem. Soc. Faraday Trans. 86 (1990) 145–149. [21] I.C. Sanchez, J. Cho, W.J. Chen, Universal response of polymers, solvents, and solutions to pressure, Macromolecules 26 (1993) 4234–4241.

JCT 2019-209