Journal of Chemical Engineering of Japan, Vol. 37, No. 12, pp. 1427–1435, 2004

Research Paper

Experimental Measurement and Model Based Inferencing of Solubility of Polyethylene in Xylene Ruchi AGARWAL, Durga PRASAD, Sunil MAITY , Kalyan GAYEN and Saibal G ANGULY Department of Chemical Engineering, Indian Institute of Technology, Kharagpur-721302, India

Keywords: Solid–Liquid Equilibrium, Laser Sensing, Polyethylene, Model Predictor Solubility of polyethylene in mixed xylene was determined experimentally under atmospheric pressure by an indigenously developed laser based technique. In this work, a PC-SAFT equation of state was used to model solid–liquid equilibrium (SLE). With the experimental SLE data available in the literature for low molecular weight n-alkanes and aromatic compounds under atmospheric pressure, the suitability of the developed SLE model based on the PC-SAFT equation of state was tested. Subsequently a sensitivity study was performed to understand the effects of different parameters that affect the solubility of polyethylene. The validated model was then used to correlate the experimentally determined solubility data for the polyethylene system.

Introduction The study of solubility of polyethylene is of great technical interest for developing and designing separation processes, such as crystallization and fractionation and solving industrial problems. In the polymer industry, deposition of polyethylene on the reactors surfaces, heat exchangers and flash drums and clogging of pipelines are some of the frequently encountered industrial problems. These industrial problems can be investigated and rectified with the proper knowledge of solid–liquid equilibrium and availability of a validated computer based model. Experimental solubility data for polyethylene systems is rarely available in the literature. The solubility of polyethylene in mixed xylene was measured experimentally using a laser based technique under atmospheric pressure. In the present work, the PC-SAFT equation of state of Gross and Sadowski (2001), was used to model solid–liquid equilibrium since it has wide applicability from low molecular weight organic compounds to highly non-ideal macro-molecular weight systems such as polymers. This equation of state requires three pure component parameters: segment number (m), segment diameter (σ ), and energy parameter (ε/ κ). Additionally PC-SAFT has adjustable solvent–solute binary interaction parameter (Kij). This model was tested from totally crystalline to partially crystalline solutes such as polymer. Using the experimental solubility data Received on July 22, 2003. Correspondence concerning this article should be addressed to S. Ganguly (E-mail address: [email protected]).

Copyright © 2004 The Society of Chemical Engineers, Japan

available in the literature for low molecular weight solutes, the model was first tested. Sensitivity analysis for the polyethylene system was performed using the tested model to understand the effects of pressure, crystallinity, melting temperature and binary interaction parameter (Kij) on solid–liquid equilibrium. The binary interaction parameter and crystallinity act as the tuning parameters for the model based predictor. Finally, a comparative study was conducted using the model based predictor and the obtained experimental data. A reasonably good match was obtained between the experimental results and the predictions. This work presents a general methodology for development of model based predictors for industrial usage and its tuning with experimental data for any general polymer and solvent system. 1.

Experimental Determination of Solubility of Polyethylene

1.1 Experimental setup A schematic diagram of the apparatus used for carrying out the experimentation is shown in Figure 1. The experimental setup consists of the following: • Flash column: The flash column is made of SS316 (100 mm inside diameter and 370 mm length). The column has four openings: 1. Inlet for nitrogen 2. Outlet for vapor 3. Thermo well for placing RTD 4. Outlet to a pressure release system The column has two diametrically opposite glass windows (50 mm diameter) near the bottom. This is provided to measure the change in intensity of the laser 1427

Fig. 1

Fig. 2

Experimental set-up for solubility measurement

Block diagram for experimental determination of solubility of polyethylene in mixed xylene

beam with increasing turbidity. A tape heater is wound across the flash column to provide heating. The heating should be slow to minimize bubble formation and hence ensure steady current in the detector. The rate of heating is controlled by a PID controller. • Optical system: The optical system in this case consists of a point laser source and a photo sensor. The photo sensor works on the principle that it senses the number of photons emitted by the laser source, which is a measure of the intensity of laser beam. The photo sensor gives the measure of the beam intensity in terms of voltage, which can be read by a computer. • Condenser system: The condenser is attached with an opening in the top of flash column to condense the solvent vapors emanating out of the flash column. A water jacket is provided above the condenser to recycle the uncondensed vapor back to the flash column. Pressure of the system is measured by the pressure transmitter, which is placed at the end of vapor line beyond the water jacket. • Pressure control system: The pressure signal from pressure transmitter is sent to the PID pressure controller to control the system pressure.

1.2 Experimental procedure The solubility of low density polyethylene in mixed xylene of analytical grade was determined under atmospheric pressure for two commercial grades of polyethylene (Agarwal, 2004). The solid–liquid equilibrium temperature or the cloud point was measured using the indigenously developed laser based technique as used by Ochi et al. (1999). A block diagram of the experimental setup along with instrumentation is shown in Figure 2. Polyethylene was weighed in a digital balance of 0.001 accuracy and mixed in a known volume of mixed xylene previously charged into the flash column. The pressure controller was given the set point at 1 bar. The laser-sensor assembly was properly aligned, so that the intensity of laser beam passing through the glass windows of the flash column falls on the sensor. Cold water was repeatedly charged in the condenser and cooling jacket assembly. The mixture was given slow and controlled heating by giving small step changes (2°C) to the temperature set point in the PID temperature controller. This ensured uniform temperature and composition in the column. Such method also helped in effective detection of the change in intensity of laser beam appearing at the formation or disappearance of turbidity. The temperature of the system was detected with an accuracy of 1°C with the help of an RTD sensor. The system temperature was increased to a maximum of 120°C after which controlled cooling was performed by successively decreasing the set point temperature of PID controller. The voltage readings of the sensor were taken for increasing and decreasing temperatures. The minimum temperature at which the intensity of laser beam remained nearly constant was marked as the cloud point. Cloud points were obtained in both heating and cooling cycles. The experiment was repeated at constant pressure of 1 bar for different weight fractions of polyethylene and for both the grades of polyethylene. 2.

Experimental Determination of Molecular Weight

Molecular weights of polyethylene samples were determined using a temperature controlled kinematic viscosity bath following the ASTM D1601 standard test method and Ostwald viscometer of ASTM D445 specification. For a given polymer–solvent system and at a given temperature, the average molecular weight of polymer can be correlated with the intrinsic viscosity of polymer solution by the modified Mark-Houwink equation given by Kurata and Stockmayer (1963). [ηi] = 0.032k(Mv/1000)a

(1)

where, k and a are empirical constants. 1428

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Polyethylene properties and xylene composition

Table 1 Properties of polyethylene Sample

Melting point [K]

Molecular weight [g/mol]

Specific gravity [—]

Crystallinity [—]

Grade 1 Grade 2

405 410

30398.9 32599.8

0.919 0.927

0.399 0.444

o-Xylene

m-Xylene

p-Xylene

0.39

0.15

0.46

Composition of mixed xylene Mole fraction [—]

The intrinsic viscosity is defined as the reduced viscosity at infinite dilution.

ηred [ηi ] = clim →0

(2 )

 η − ηs   c  ηs 

[ηred ] = 

(3)

The values of empirical constants determined experimentally from known molecular weight polyethylene samples in mixed xylene were found to be k = 4.12274 and a = 0.714757. The specific gravity readings of polymer samples were determined using the specific gravity measurement bottles. The molecular weight, melting point and specific gravity of the two polyethylene samples and the composition of mixed xylene are tabulated in Table 1. 3.

Modeling of Solid–Liquid Equilibrium

The fundamental principle of any phase equilibrium calculation is that the fugacity of any component is equal in all the phases under equilibrium condition. With this fundamental principle, Pan and Radosz (1999) developed the solid–liquid equilibrium model that is applicable to totally crystalline solutes. φ L X L ln  2 0 2  φ

  ∆Hm  Tm  ∆v − 1 + P − P sat   = −  RT  RTm  T  

(

)

( 4)

where, φ0 is the fugacity coefficient of pure sub-cooled liquid solute at constant T and P. φ 2L is the fugacity coefficient of solute in a solution, X 2L is the equilibrium solubility of solute, Psat is the saturated-vapor pressure of solute at its melting temperature (Tm). ∆v is the volume difference of liquid and solid solute defined as ∆v = vL – vS, and ∆Hm is the enthalpy of fusion of solute. Subsequently the author extended this equation for partially crystalline solutes such as polyVOL. 37 NO. 12 2004

mer with the hypothesis of Harismiadis and Tassios (1996), who assume that the logarithm of the ratio of fugacities is proportional to crystallinity (C). The crystallinity fraction of polymer is the fraction of crystalline substances present in the sample. The amorphous polymers will have zero crystallinity. φ L X L   ∆H T ∆vP  ln  P 0 P  = −  u  m − 1 + Cu  RT   RTm  T  φP 

(5)

where, ∆Hu is the enthalpy of melting per mole of crystal unit. For the ethyl unit, ∆Hu = 8.22 kJ/mol, as reported by Van Krevelen (1990). The polymer-volume change, ∆v, is determined from the densities of an amorphous polymer, ρa, and a crystalline polymer, ρc ∆v =

1 1 − ρa ρ c

(6 )

For polyethylene, ρa = 0.853 g/cm3 and ρc = 1.04 g/cm3. The crystallinity of the polymer samples was calculated using the following equation  ρ   ρ − ρa  C =  c    ρ   ρ c − ρa 

(7)

where, ρ = specific gravity of polymer. The value of crystallinity (C) of the two samples is tabulated in Table 1. Solid crystalline normal alkanes such as noctacosane and n-dotriacontane exhibit a solid–solid (ss) phase transition a few degrees below its melting point. Since the two solid phases are in a state of thermodynamic equilibrium, the effect of the solid–solid phase transition is included as follows: φ L X L   ∆H  ∆Hss  Tss T ln  2 0 2  = −  m  m − 1 + − 1      RTss T  RTm T   φ 

(8)

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Table 2

PC-SAFT parameters of organic solutes and solvents

Hydrocarbon Solutes n-Dodecane (C1 2 ) n-Hexadecane (C1 6 ) n-Octadecane (C1 8 ) n-Octacosane (C2 8 ) n-Dotriacontane (C3 2 ) Biphenyl Solvents n-Hexane n-Heptane n-Decane Benzene m-Xylene o-Xylene p-Xylene

Segment No., m [—]

Segment diameter, σ [Å]

Energy parameter, ε/ κ [K]

5.3060 6.6485 7.3271 10.3622 11.835 3.8877

3.8959 3.9592 3.9668 4.0217 4.0217 3.8151

249.21 254.70 256.20 252.0 252.0 327.42

3.0576 3.4831 4.6627 2.4653 3.1861 3.1362 3.1723

3.7983 3.8049 3.8384 3.6478 3.7563 3.7600 3.7781

236.77 238.40 243.87 287.35 283.98 291.05 283.77

where, Tss is the ss transition temperature and ∆H ss is the enthalpy of the ss transition. The fugacity coefficients of solute appeared in the above three equations are calculated by the PC-SAFT equation of state. For the correlation and prediction of phase equilibrium in macromolecular systems, the equations of state for chain molecules have been successfully used for more than two decades. Gross and Sadowski (2001) developed a perturbed-chain SAFT equation of state based on a perturbation theory. In this model, molecules are considered as chains composed of spherical segments. According to the perturbation theory, the potential function can be divided into repulsion and attraction components. Attraction may be due to dispersion or association. In the present work, only dispersion based attraction is taken into consideration (Abbas et al., 2004). Special types of forces like hydrogen bonding and dipole–dipole interaction have not been considered. The model is applicable to real chain molecules of any length, from spheres to polymers and can be used to calculate density, vapor pressure, and caloric properties (Mukherjee et al., 2002; Mukherjee, 2003). 4.

Tuning of the SLE Model Using Existing Solubility Data

The suitability of the developed composite model was tested by using the experimental solubility data for low molecular weight organic compounds collected from the existing literature. PC-SAFT parameters of these solvents and solutes are listed in Table 2. 4.1 Solid–liquid equilibrium of n-alkanes Figure 3 shows the solubility of n-dodecane and n-hexadecane in n-hexane solvent, on the basis of ex-

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Fig. 3

SLE for n-dodecane and n-hexadecane in n-hexane solvent at 1 bar

perimental data taken from Hoerr and Harwood (1951). These data could be predicted by a PC-SAFT model with Kij = 0.0. Solid–solid transition and melting properties are tabulated in Table 3. Figure 4 represents the solubility of n-octadecane and n-dotriacontane in n-heptane solvent. The corresponding experimental data were taken from Chang et al. (1983). The comparisons of experimental data with model-based predictions corresponding to K ij = 0.0 showed that the model predictions could be used only with a small error. 4.2 Solid–liquid equilibrium of aromatic compounds McLaughlin and Zainal (1959) studied the solubility of biphenyl in benzene for higher mole fraction range. The mode-based predictions have been compared with their experimental data as shown in Figure 5. The comparative results showed that the model JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

Table 3

Solid–solid transition and melting properties

Hydrocarbon

Tss [K]

n-Dodecane (C1 2 ) n-Hexadecane (C1 6 ) n-Octadecane (C1 8 ) n-Octacosane (C2 8 ) n-Dotriacontane (C3 2 ) Biphenyl

331.2 338.9

∆Hss [J/mol]

35447 42700

Tm [K]

∆Hm [J/mol]

263.6 291.2 301.1 334.4 342.1 342.1

36977 53563 59400 64658 76000 18732

Molar volume [cm3 /mol] of n-octacosane (C2 8 ) at temperature T [K] vL = 0.1238T + 365.588 vS = 0.11828T + 381.623

Table 4 Polyethylene LDPE HDPE

Fig. 4

PC-SAFT parameters of polyethylene

Segment No., m/M [mol/g]

Segment diameter, σ [Å]

Energy parameter, ε/ κ [K]

0.0263 0.0263

4.0217 4.0217

249.5 252.0

SLE for n-octadecane and n-dotriacontane in nheptane at 1 bar

Fig. 6

The effect of pressure on the solubility of polyethylene in m-xylene

predictions are in good agreement with the experimental data. 5.

Fig. 5

SLE for biphenyl in benzene at 1 bar

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Sensitivity Analysis for a Polyethylene–Xylene System

A sensitivity study was performed using the tested SLE model as used for low molecular weight systems, to understand the effects of crystallinity, melting temperature, adjustable binary interaction parameter (Kij) and pressure on solid–liquid equilibrium of polyethylene in m-xylene. The PC-SAFT parameters of polyethylene used by Gross and Sadowski (2002), for high-pressure phase equilibrium calculation, was used for this study as listed in Table 4.

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Fig. 7

Fig. 8

The effect of crystallinity (C) on the solubility of polyethylene in m-xylene at 1 bar

The effect of melting point (T m) on the solubility of polyethylene in m-xylene

Fig. 10 Variation in the intensity of laser beam with an increase and a decrease in temperature for 0.054 weight fraction of polyethylene (30398.9)

mous change in pressure can decrease the solubility of polyethylene to a great extent sufficient to cause industrial problems like clogging of pipelines due to the deposition of polyethylene. With an increase in crystallinity (C), the solubility of polyethylene decreases at a fixed temperature and pressure as shown in Figure 7. The influence of crystallinity on the solubility has been found to be substantial during this study. With an increase in the melting point, the solubility of polyethylene decreases at a fixed temperature, pressure, Kij, and crystallinity as shown in Figure 8. As shown in Figure 9, the effect of binary interaction parameter, Kij on the solubility of polyethylene was observed to be very small for low weight fractions of polyethylene. From the figures presented in this study, it was observed that the solute is totally soluble in the solvent at the melting point of solute. 6.

Fig. 9

The effect of Kij on the solubility of polyethylene in m-xylene

At a fixed temperature, the solubility decreases with the increase in pressure as shown in Figure 6. But the effect of pressure on solubility was found to be substantial only for large change in pressure. Enor-

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Interpretation of Experimental Results for the Cloud Point

Figures 10 and 11 show the variation in laser beam intensity by increasing and decreasing temperature cycles for 0.054 and 0.11 weight fraction respectively of Grade 1 polyethylene. Mixed xylene and polyethylene pellets were present in the solution at the start of experiment. With increasing temperature, initially the polyethylene got dispersed in the xylene phase and the turbidity of the solution increased. This caused the decrease in intensity of the laser beam which was manifested by the decrease in the voltage reading of the sensor. After a point of minimum intensity was reached, with further increase in temperature the polyethylene started getting dissolved into the xylene and the turbidity of the

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Table 5

SN

Weight fraction of polyethylene [—]

Cloud point [K]

1 2 3 4 5 6

0.0149 0.0540 0.110 0.25 0.36 0.42

370 375 379 378 379 381

Table 6 Fig. 11 Variation in the intensity of laser beam with an increase and a decrease in temperature for 0.11 weight fraction of polyethylene (30398.9)

Fig. 12 Variation in the cloud point with an increase in weight fraction of polyethylene (30398.9)

solution decreased. The intensity of laser beam started to increase until a minimum temperature was attained at which all of the polyethylene had dissolved in xylene. This was the saturation solubility temperature or the cloud point. Beyond this point, the laser beam intensity remained nearly constant. In the decreasing temperature cycle, as the temperature of the system was lowered by controlled cooling, a cloud point was attained after which the intensity of laser beam started decreasing because of the precipitation of polyethylene from the solution. A marked shift in the minimum intensity point for the increasing and decreasing temperature cycle was observed. This may be attributed to the change in the physical structure of the polyethylene after precipitation. Beyond the minimum intensity point in the cooling cycle, the laser beam intensity started to increase

VOL. 37 NO. 12 2004

The experimental cloud point for Grade 1 polyethylene in xylene

The experimental cloud point for Grade 2 polyethylene in xylene

SN

Weight fraction of polyethylene [—]

Cloud point [K]

1 2 3 4 5 6

0.0149 0.0540 0.110 0.25 0.36 0.42

376 377 380 380 382 384

but was unable to reach its starting value. This again may be due to the fact that the precipitated polyethylene was not recovered in its original physical structure even after substantial cooling of the system. Figure 12 shows a study of the variation of cloud point with an increase in weight fraction of Grade 1 polyethylene for increasing temperature cycle. There is an increase in the cloud point of the system with an increase in polyethylene weight fraction. This indicates that a higher temperature is required for the increased amount of solute to dissolve in the solvent. The results of the experiments conducted for determination of the cloud points for different weight fractions of the two grades of polyethylene are shown in Tables 5 and 6. 7.

Comparative Study of Experimental Data with a Model Based Predictor

The solubility data for polyethylene having a molecular weight of 17000 g/mol and a melting point of 387.5 K and a crystallinity of 0.8 in m-xylene at atmospheric pressure is available in the existing literature, Pan and Radosz (1999). However, tuning of the PC-SAFT based predictor for real life systems has rarely been studied in the literature. The general methodology proposed in this work was used to compare the model-based predictions with experimental data. Figure 13 shows the comparative study between the experimental solubility data of the present work and the model based predictions for polyethylene

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Conclusion

Fig. 13 Solubility of polyethylene (PE32599.8) in xylene at 1 bar

The solubility of low density polyethylene in mixed xylene was measured under atmospheric pressure using an indigenously developed laser based technique. A computer based general methodology developed using the PC-SAFT equation of state was used to predict the solid–liquid equilibrium of real solutes and polymers under atmospheric pressure. For a given molecular weight and melting temperature of solute, the model predictions at atmospheric pressure were obtained by tuning the adjustable model parameters like crystallinity and a binary interaction parameter (Kij), with the experimental solubility data. A comparison of the model predictions with the experimental data for low molecular weight organic solutes and polymers showed that the developed model could be used to predict the solid–liquid equilibrium. Acknowledgment Financial assistance from MHRD, project no. F27-1/2002 TS.V dated March 19, 2002 is thankfully acknowledged. Nomenclature C = c = K ij = M = Mv = m = P = PC-SAFT =

Fig. 14 Solubility of polyethylene (PE30398.9) in xylene at 1 bar

(32599.8) for different values of crystallinity (C) at 1 bar. The model predictions were obtained using constant values for the following parameters: Melting temperature = 410 K Binary interaction parameter Kij = 0 The comparative results indicate that the model predictions match the experimental data closely with crystallinity (C) = 0.8. The experimental solubility data for polyethylene (30398.9) is compared with the model predictions at 1 bar in Figure 14, using the following model parameters: Melting temperature = 405 K Binary interaction parameter Kij = 0 Crystallinity (C) = 0.8

PE P sat

= =

R SAFT SLE SS T u v X

= = = = = = = =

ε ∆Η ∆Ηu ∆ν η ηi ηred ηs κ ρ ρa ρc σ φ φ0

= = = = = = = = = = = = = = =

m = P =

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crystallinity [—] concentration [g/dl] binary interaction parameter [—] molar mass [g/mol] viscosity average molecular weight [g/mol] number of segments per chain [—] pressure [bar] perturbed-chain statistical associating fluid theory [—] polyethylene [—] saturated pressure of solute at a melting temperature [bar] gas constant [J mol –1 K –1] statistical associating fluid theory [—] solid–liquid equilibrium [—] solid–solid [—] temperature [K] number of ethyl units in the backbone [—] molar volume [cm3/mol] mole fraction [—] depth of pair potential [J] molar enthalpy change [kJ/mol] enthalpy of melting per crystal unit [kJ/mol] molar volume change [cm3/mol] solution viscosity [dl/g] intrinsic viscosity [dl/g] reduced viscosity [—] solvent viscosity [dl/g] Boltzmann constant [J/K] density [mol/ml] density of completely amorphous solute [g/cm 3] density of completely crystalline solute [g/cm 3] segment diameter [Å] fugacity coefficient [—] fugacity coefficient of pure sub cooled liquid solute at given temperature and pressure [—]

melting condition polymer

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ss 2

= =

L = S = sat =

solid–solid phase transition solute

liquid phase solid phase property at saturation condition

Literature Cited Abbas, S., R. Mukherjee, S. De and S. Ganguly; “Real-Time Inferencing of Solid–Liquid Phase Equilibria in Solution Polymerization of Polyethylene,” Chem. Eng. Process., 43, 1449– 1458 (2004) Agarwal, R.; Experimental Determination and Modeling of Solid– Liquid Equilibrium in Polyethylene Systems, M.Tech. Thesis, Indian Institute of Technology, India (2004) Chang, S. S., J. R. Maurey and W. J. Pummer; “Solubilities of Two n-Alkanes in Various Solvents,” J. Chem. Eng. Data, 28, 187– 189 (1983) Gross, J. and G. Sadowski; “Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules,” Ind. Eng. Chem. Res., 40, 1244–1260 (2001) Gross, J. and G. Sadowski; “Modeling Polymer Systems Using the Perturbed-Chain Statistical Associating Fluid Theory Equation of State,” Ind. Eng. Chem. Res., 41, 1084–1093 (2002)

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Harismiadis, V. I. and D. P. Tassios; “Solid–Liquid–Liquid Equilibria in Polymer Solutions,” Ind. Eng. Chem. Res., 35, 4667–4681 (1996) Hoerr, C. W. and H. J. Harwood; “Solubilities of High-MolecularWeight Aliphatic Compounds in n-Hexane,” J. Org. Chem., 16, 779–791 (1951) Kurata, M. and W. H. Stockmayer; “Intrinsic Viscosities and Unperturbed Dimensions of Long Chain Molecules,” Fortschr. Hochpolym. Forsch., 3, 196 (1963) McLaughlin, E. and H. A. Zainal; “The Solubility Behavior of Aromatic Hydrocarbons in Benzene,” J. Chem. Soc., 863 (1959) Mukherjee, R.; Simulation of a LLDPE Reactor for Real-Time Application, M.Tech. Thesis, Indian Institute of Technology, India (2003) Mukherjee, R., S. Abbas, S. De, S. Ganguly, B. S. G. Ramprasad and K. Sen; “Simulation of a Steady State Non-Isothermal Polymerization Reactor Using Ziegler–Natta Catalyst,” Proceedings, CHEMCON, Hyderabad, India (2002) Ochi, K., T. Saito and K. Kojima; “Determination of Solubilities of Polymers in Solvents by a Laser Scattering Technique,” Fluid Phase Equilib., 158–160, 847–855 (1999) Pan, C. and M. Radosz; “Modeling of Solid–Liquid Equilibria in Naphthalene, Normal-Alkane and Polyethylene Solutions,” Fluid Phase Equilib., 155, 57–73 (1999) Van Krevelen, D. W.; Properties of Polymers, p. 120, Elsevier, Amsterdam, the Netherlands (1990)

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