Mechanical characteristics of modified unsaturated polyester resins

0 downloads 0 Views 143KB Size Report
properties due to their liquid crystalline behaviour. Here we study ... Modified unsaturated polyesters were synthesized by depolymerizing PET with. DEG, and ...
Polymer International

Polym Int 51:183±189 (2002) DOI: 10.1002/pi.818

Mechanical characteristics of modified unsaturated polyester resins derived from poly(ethylene terephthalate) waste Medhat S Farahat1,2* 1

University of Alabama, Center for Materials for Information Technology, Box 870209, Tuscaloosa, Alabama 35487-0209, USA Egyptian Petroleum Research Institute (EPRI), Nasr City 11727, Cairo, Egypt

2

Abstract: The depolymerization of poly(ethylene terephthalate) (PET), by an alcoholysis reaction is an easy operation and gives prospects for the utilization of wastes. PET waste was ®rst depolymerized by glycolysis reaction at three different molar ratios of diethylene glycol (DEG), in the presence of manganese acetate as a transesteri®cation catalyst. Copolyesters of PET modi®ed with varied mole ratios of p-hydroxybenzoic acid (PHBA) were reported to exhibit excellent mechanical and chemical properties due to their liquid crystalline behaviour. Here we study the effect of incorporating (PHBA) units into the building structures of different unsaturated polyesters synthesized originally from glycolysed PET waste. Modi®ed unsaturated polyesters were synthesized by depolymerizing PET with DEG, and the obtained oligoesters were reacted with PHBA and maleic anhydride (MA). The molar ratio of the added PHBA was varied to investigate its effect on the mechanical characteristics of these modi®ed unsaturated polyesters. The data obtained reveal that increasing the molar ratio of PHBA within the studied range of concentrations leads to a pronounced improvement in the mechanical characteristics, which is represented mainly by the values of/maximum compression strength (smax) and Young's modulus (EY). # 2002 Society of Chemical Industry

Keywords: modi®ed unsaturated polyesters; p-hydroxybenzoic acid; recycling of poly(ethylene terephthalate); glycolysis reaction; mechanical characteristics

INTRODUCTION

Poly(ethylene terephthalate) (PET) is a thermoplastic polyester which is produced in considerable amounts because it ®nds applications in the textile industry, high strength ®bres, photographic ®lms and soft drink bottles. The wastes from PET spinning are not suitable for a second drawing. Because of the new regulations which have been set up at the national, European and world levels, the utilization of the polymer waste becomes a priority. One possible approach to its reuse consists of transforming it into oligoester polyols by alcholytic destruction.1±7 It is well known8±16 that glycolysis of PET can be carried out in the presence of a transesteri®cation catalyst to produce oligomeric polyols. These polyols are used as starting materials for other industries as are polyurethanes (PU), unsaturated polyesters4 and dialkyl terephthalate plasticizers.17 Copolyesters of PET and various mole ratios of p-acetoxybenzoic acid (PABA), commonly referred to as PETA/x-PABA, where x denotes to the mole ratio of PABA in the copolyester, were reported to exhibit excellent mechanical and chemical resistance properties due to their liquid crystalline

character.18±22 In this paper, PET was depolymerized by diethylene glycol (DEG) at three different molar ratios of DEG:PET. The glycolysed products thus obtained were subjected to polyesteri®cation reactions with maleic anhydride (MA) and different molar ratios of p-hydroxybenzoic acid (PHBA). PHBA was added at different molar ratios to investigate its effect on improving the mechanical properties of the cured polyesters.

EXPERIMENTAL Glycolysis of PET waste

To study the improvement in the mechanical properties as a result of incorporating p-hydroxybenzoic acid units into oligomers containing terephthalate repeating units, twelve different unsaturated polyester resins were synthesized from PET waste and co-reacted with various molar ratios of PHBA.23 DEG, manganese acetate (a transesteri®cation catalyst), MA and PHBA were reagent grade chemicals obtained from Aldrich Chemical Co, and used without further puri®cation. PET waste obtained from beverage bottles was

* Correspondence to: Medhat S Farahat, University of Alabama, Center for Materials for Information Technology, Box 870209, Tuscaloosa, Alabama 35487-0209, USA E-mail: [email protected] (Received 24 January 2001; revised version received 31 August 2001; accepted 25 September 2001)

# 2002 Society of Chemical Industry. Polym Int 0959±8103/2002/$30.00

183

M S Farahat

collected, washed thoroughly, dried and chopped into very small pieces. Three different glycolysed products of PET (glycolysates), denoted GLY1, GLY2 and GLY3, were prepared by applying three different molar ratios of DEG:PET, namely 1.15, 1.54 and 1.94. Calculations are based on the molecular weights of DEG and repeating unit of PET. The glycolysis reaction was carried out in a continuously stirred three-necked ¯ask equipped with a condenser. The temperature of the reaction was kept constant at 200±220 °C for 4 hours.5,24±27 After completion of the glycolysis reaction, the product was cooled to room temperature, dissolved in a suitable amount of an organic solvent (eg CH2Cl2, CCl4 or benzene) and shaken vigorously with an equal amount of distilled water to remove any unreacted glycol and remaining catalyst.7 After phase separation between the two layers, the organic layer (oligomer dissolved in the organic solvent), was collected and washed several times with distilled water using the same procedure. Finally, puri®ed oligomers were obtained after removal of the solvent by rotary evaporation. Molecular weight determination

The glycolysates GLY1, GLY2 and GLY3 obtained from the PET glycolysis process, were puri®ed and tested for hydroxyl values (B), expressed as mg KOH/g sample.28±31 The number average molecular weights hMni could then be calculated by substitution into eqn (1).32 hMn i ˆ …2  56:1  1000†=B

…1†

Synthesis of modified unsaturated polyester resins (UP)

The puri®ed glycolysates GLY1±GLY3 were subjected to a non-catalysed melt polyesteri®cation reaction with MA. GLY1±GLY3 were reacted with MA at the molar ratio of 1:1; calculations were based on the values of hMni, obtained from eqn (1). Different molar ratios of PHBA, namely 0.0, 0.071, 0.142 and 0.213 moles per mole of the total MA added, were added to modify these unsaturated polyesters. The number average molecular weights hMni of the synthesized unsaturated polyesters (UP1±UP12) were determined by end-group analysis.32,33 Each UP was analysed for both acid number (A) and hydroxyl value (B) in mg KOH/g resin, and hMni values were calculated by substitution into eqn (2).32 hMn i ˆ …2  56:1  1000†=…A ‡ B† 1

H NMR and

13

…2†

C NMR spectroscopy For the purpose of investigating the of incorporation of p-oxybenzoate units into the backbones of the modi®ed polyesters, samples UP9±UP12 were analysed using both 1H NMR and 13C NMR spectra. 1H NMR and 13C NMR spectra were recorded using a Bruker 184

AM360 NMR instrument. 1H NMR spectra were measured at 360.13 MHz. Samples UP9±UP12 were selected for both NMR analyses because they contain the highest concentrations of PHBA. For the measurements, 2.0 mg of the sample was dissolved in 0.5 ml of deuterated tri¯uoroacetic acid (CF3COOD) at 25.0 °C. X-ray diffraction

X-ray diffraction patterns were obtained with Rigaku Geiger X-ray diffractometer, using CuKa radiation (40 kV, 40 mA). Samples UP9±UP12 were tested for X-ray diffraction to investigate any changes in crystallinity resulting from importing PHBA into polyester chains. Test samples were prepared by putting a thin layer coating of polyester onto the surface of an [001] silicon wafer and then measuring the X-ray diffraction pattern. Mechanical testing

The mechanical characteristics of the cured polyesters under investigation, represented by Young's modulus (EY) and maximum compressive strength (smax) were measured by a Lloyd Instruments Ltd universal testing machine (capacity 10 tons). Testing of mechanical properties was carried out according to ASTM method D695-44T.

RESULTS AND DISCUSSION

Thermotropic liquid crystalline polymers (LCP) are a relatively new class of high performance materials which combine excellent thermal stability and mechanical properties.34 Weinkauf and Paul35 have studied extensively the gas transport properties of liquid crystalline poly(ethylene terephthalate-co-p-oxybenzoate) and attributed their excellent properties to the high liquid crystalline order of such a class of copolymers. It is well known35±38 that the incorporation of PHBA units into polymer chains containing terephthalate repeating units increases the chain order and hence leads to an increase in crystallinity and mechanical properties of these copolymers. However, to the author's knowledge, no work has been reported in the literature on the synthsis of modi®ed copolyesters, initially derived from PET waste, containing both p-oxybenzoate and terephthalate repeating units. In an earlier publication,23 we reported on the improvement of the curing characteristics of these modi®ed unsaturated polyesters. In the present work, we are pursuing our investigation on the improvement in mechanical properties resulting from the incorporation of PHBA into the backbones of these modi®ed polyesters. For this purpose, twelve different unsaturated polyesters were synthesized from three different glycolysed products of PET (GLY1±GLY3). The molar ratios of all ingredients involved in both steps of glycolysis and polyesteri®cation reactions, calculated acid numbers, hydroxyl values and number average molecular weights hMni of GLY1±GLY3 and Polym Int 51:183±189 (2002)

Mechanical characteristics of unsaturated polyester resins Table 1. Characterizations and constituents involved in the preparations of UP1 –UP12

Code for [OH] value, Polyester DEG: PET glycolysate (mg KOH/ hMni  10 resin code molar ratio obtained g) (eqn 1)

2

Moles of MA added/100 g glycolysate

Moles of PHBA added

Acid [OH] value number (mg KOH/ (mg KOH/g) g)

hMni  10 (eqn2)

UP1 UP2 UP3 UP4

1.15

GLY1

124.1

9.041

0.101

0.000 0.007 0.014 0.022

20.2 18.1 18.6 19.1

18.6 19.7 17.5 15.5

2.890 2.971 3.110 3.240

UP5 UP6 UP7 UP8

1.54

GLY2

163.2

6.875

0.132

0.000 0.009 0.019 0.028

18.2 17.3 15.7 14.2

23.1 22.8 22.1 21.8

2.720 2.800 2.964 3.121

UP9 UP10 UP11 UP12

1.92

GLY3

234.4

4.787

0.190

0.000 0.013 0.027 0.041

16.9 15.2 13.9 12.4

27.1 26.5 26.9 26.2

2.550 2.690 2.751 2.910

the synthesized unsaturated polyesters UP1±UP12 are shown in Table 1. The modi®ed unsaturated polyesters UP1±UP12 were dissolved in 40% (w/w) of styrene monomer and cured in a thermostatted water bath at 35 °C. The concentration of the free radical initiator, methyl ethyl ketone peroxide (MEKP), was 2% (w/w) and that of the promoter or accelerator, cobalt octoate, was 0.2% (w/w) with respect to the total weight of the styrenated polyester samples.23 Polymer structure determined by 1H NMR spectroscopy

It was reported that, for the sequence distribution of the PET/60-PABA copolyesters measured by 1H NMR spectrophotometry,39 the PABA units have a random distribution in the copolymer.40,41 Nicely et al 42 also studied the sequence distribution of the PET/ 60-PABA copolyesters by 1H NMR spectroscopy.

3

Kang and Ha36 carried out 1H NMR spectrophotometry on a series of poly(ethylene terephthalate-co-poxybenzoate) copolyesters with varying concentrations of PABA. According to the reported results,36,39±42 1 H NMR spectrophotometry of these copolyesters gave a direct assessment of the probability of the PABA unit being bonded to another PABA or to a PET unit.42 Figure 1 shows the global 1H NMR spectra for samples UP9±UP12. Figure 2 focuses on the characteristic peaks occurring between 7.70 and 8.20 ppm in Fig 1. The work done by these authors,36,39±42 was carried out on more de®nite systems composed of PET and x-PABA whereas the current work is more complicated, where the system is composed of PET repeat units/DEG/MA and PHBA. However, characteristic 1H NMR peaks at 7.85 and 7.80 ppm were detected and taken as evidence for the probability of PHBA±PHBA and PHBA±PET linkages respectively.

Figure 1. Global 1H NMR spectra for samples UP9–UP12 (360.13MHz, solvent CF3COOD).

Polym Int 51:183±189 (2002)

185

M S Farahat

Figure 2. Characteristic peak change in 1H NMR spectra for samples UP9–UP12.

As can be seen in Fig 2, UP9 showed one 1H NMR peak at 7.83 ppm. This peak, corresponding to a PETaliphatic unit, was gradually replaced by the other two peaks occurring at 7.85 ppm and 7.80 ppm, which correspond to PHBA±PHBA and PHBA±PET linkages, respectively. Also, the intensity of the peak occurring at 7.80 ppm was found to increase while that occurring at 7.85 ppm decreases as the concentration of PHBA increases. This indicates that PHBA±PET units become more ordered than PHBA±PHBA units as the concentration of PHBA increases from UP10 to UP12. Polymer structure determined by spectroscopy 13

13

C NMR

C NMR analysis was carried out for samples UP9±

UP12 dissolved in deuterated tri¯uoroacetic acid. Figure 3 shows the global 13C NMR spectra with the possible interpretation of the characteristic peaks. Peaks denoted a, b, c and d, which occur at 122.3 ppm, 119.2 ppm, 116.2 ppm and 112.8 ppm, respectively, are assigned to the four different C atoms in the aromatic rings of the terephthalate units. The other peaks, which appeared signi®cantly in the case of UP12, denoted by e, f, g and h, and occurring at 121.2 ppm, 118.2 ppm, 114.9 ppm and 111.7 ppm are assigned to the C atoms in the aromatic ring of poxybenzoate units. Peak j, which appears at 133.0 ppm was assigned to the maleate oli®nc C atoms. The new peak k, occurring at 132.1 ppm, which appeared obviously in the case of UP12, was assigned together with peak j to the two dissimilar oli®nic C atoms of maleate repeating unit. The occurrence of peak k is supposed on the basis that there is a possibility that the carboxylic groups of the maleate units could react with either the aliphatic (CH2ÐOH), or with the phenolic (ÐOH) group of p-oxybenzoate units. X-ray diffraction patterns

The physical and mechanical properties of polymers are profoundly affected by the change in crystallinity. In the current work, the change in crystallinity was recorded for samples UP9±UP12, because they contain the highest concentrations of PHBA. The diffraction patterns were recorded at the Bragg angles between 2y = 10° and 2y = 30°, as a measure of the change in the degree of crystallinity of samples, using wide angle X-ray spectrometry. As can be seen in Fig 4, the pattern and the relative intensities became gradually narrower upon changing from UP9 to UP12. The estimated peak width, recorded at its mid-height, was found to be 25 mm, 21 mm, 18 mm and 17 mm for samples UP9, UP10, UP11 and UP12, respectively.

Figure 3. Global 13C NMR spectra for samples UP9–UP12 (360.13MHz, solvent CF3COOD).

186

Polym Int 51:183±189 (2002)

Mechanical characteristics of unsaturated polyester resins

samples, which contain a gradual increase of PHBA molar concentrations, represent the modi®ed unsaturated polyesters. Young’s modulus (EY)

Figure 4. XRD patterns for samples UP9–UP12.

This gradual decrease in the peak width of the XRD pattern, indicates that there is some degree of crystallinity imparted as a result of increasing the molar concentration of PHBA from samples UP9 to UP12. Measurement of the mechanical properties

Cured polyesters were molded in the form of rods. These rods were cut into straight cylinders with de®nite dimensions according to the ASTM method D695-44T. Samples were tested for the maximum compressive strength (smax) and Young's modulus (EY). The measured values for EY (MPa)  10 1 and smax (MPa) were tabulated and are shown in Table 2. Each group of tested polyesters was classi®ed according to the type of glycolysate (GLY1±GLY3) incorporated into their preparations. Samples UP1, UP5 and UP9 in each set of polyesters, represent the unmodi®ed unsaturated polyesters, while the rest of the

Table 2. Mechanical characteristics of the cured polyesters UP1–UP12

Polym Int 51:183±189 (2002)

Careful inspection of the values of Young's modulus, EY (MPa) shown in Table 2 and the molar concentrations of the ingredients incorporated into the syntheses of UP1±UP12 shown in Table 1, reveals that the values of EY increase linearly as the concentrations of both MA and PHBA are increased. The increase in the values of EY seen upon increasing the molar concentrations of MA can be attributed to the increase in the number of curable double bonds in the polyester backbones, thus greatly increasing the crosslink density. This resulted in the cured polyesters being harder and more resistant to deformation upon the application of external stresses. The increase in PHBA concentration led to an increase of ordered domains which also increases the resistance of the cured polyesters to external stresses. Figure 5 shows the graphical presentation of the variation in Young's modulus, EY (MPa) with the variation in the chemical composition of the cured polyesters UP1±UP12. Maximum compressive strength (smax)

It has been indicated before43 that copolymers of PET and PHBA show extremely high compression modulus values due to the increased crystallinity of such copolymers. The modulus values and crystallinity of these copolymers were found to be increased by increasing the molar ratio of PHBA.44±50 As noticed by comparing the results of maximum compressive strength smax, shown numerically in Table 2 and graphically in Fig 6, the gradual increase in PHBA molar concentrations, among all the synthesized unsaturated polyesters, led to an increase of smax. This increase in smax with the increase of PHBA molar ratios is linear in the case of the two sets of cured polyesters UP1±UP4 and UP5±UP8. These values decreased to a lower level in the case of the last set of cured polyesters UP9±UP12. The linear increase in the values of smax in the case

Unsaturated polyester resin code

Code of glycolysate incorporated

UP1 UP2 UP3 UP4

GLY1

129.2 148.3 157.1 172.4

147.1 159.1 170.5 180.6

UP5 UP6 UP7 UP8

GLY2

165.1 180.7 206.1 225.1

166.9 183.7 215.0 222.9

UP9 UP10 UP11 UP12

GLY3

235.0 241.1 245.3 251.4

113.3 121.6 127.3 138.2

EY (MPa)  10

1

smax (MPa)

187

M S Farahat

Figure 5. Variation of Young’s modulus (EY) with the chemical composition of the cured polyesters.

of UP1±UP8 can be attributed to the increase of the chain order and the more packed structures imparted as a result of increasing the molar ratio of PHBA and to the increased crosslink density imparted by increasing the MA concentration. The net effect is that the mechanical strength was signi®cantly improved as a result of the increased chain order and more chemically bonded structures. However, the further increase of MA molar concentration, as in the case of the last series of cured polyesters UP9±UP12, led to the decrease in the values of smax. The effect of increasing MA molar concentrations led to gradual improvement in smax up to a certain value. The concentrations of MA which gave the best results for smax were 0.101 mol and 0.132 mol/100 g glycolysate. As the molar concentration of MA was increased to 0.190 mol/100 g glycolysate, the value of smax decreased because the crosslink density increased to the extent that the cured polyesters became brittle, causing the mechanical properties to decrease.51 The

gradual increase in the values of smax for samples UP9 to UP12 can be attributed to the separation introduced by PHBA units between curable double bonds. Thus the crosslink density in the cured polyesters became slightly decreased as the result of these separations between crosslinks and samples became less brittle with increased concentrations of PHBA.

CONCLUSIONS

Modi®cation of the mechanical properties of the cured unsaturated polyesters, initially derived from PET waste, could be achieved successfully by the incorporation of PHBA into the polyester backbones. The values of maximum compressive strength smax and Young's modulus EY increased linearly with the increase of molar concentrations of both PHBA and MA. However, values of smax decreased at the higher concentrations of MA.

Figure 6. Variation of the maximum compressive strength smax with the chemical composition of the cured polyesters.

188

Polym Int 51:183±189 (2002)

Mechanical characteristics of unsaturated polyester resins

ACKNOWLEDGEMENTS

The author wishes to express his sincere gratitude to the Chemistry Department at the University of Alabama, Tuscaloosa, USA, for offering all the facilities required in doing most of the experimental work, especially the 13C NMR, 1H NMR spectroscopy and X-ray diffraction patterns. The author also wishes to thank Dr Rainer Schad and Dr Paul Evans for carrying out and interpreting the XRD analyses. REFERENCES 1 2 3 4

Vaidya UR and Nadkarmi VM, J Appl Polym Sci 34:235 (1987). Vaidya UR and Nadkarmi VM, Ind Eng Chem Res 26:194 (1987). Vaidya UR and Nadkarmi VM, J Appl Polym Sci 35:775 (1988). Vaidya UR and Nadkarmi VM, Polym Mater Sci Eng 63:1029 (1990). 5 Halacheva N and Novakov P, Polymer 36:867 (1995). 6 Hubert F, Duran G and Tersac G, J Appl Polym Sci 72:829 (1999). 7 Chen J-W and Chen L-W, J Appl Polym Sci 73:35 (1999). 8 Heinze J, Ramm H and Richardt H, German patent 1247291 (1967). 9 Gruschke H and Mayer M, German patent 1290929 (1969). 10 Miura K, Kagiya Y and Ichikawa T, Japanese patent 6823449 (1968). 11 Stevenson GM, German patent 1803929 (1969). 12 Etienne Y and Soulas R, French patent 1563765 (1969). 13 Siclari F, Ruta D and Cauzzi F, Italian patent 729187 (1966). 14 Ligorati F, Aglietti G and Nova VE, German patent 2158560 (1972). 15 Mueller W, Groeger C, Schmidt W and Strobel L, German patent 92801 (1972). 16 Hemmi H, Nagashima H, Kimura Y, Teresake L and Satani M, Japanese patent 7362732. 17 Dupont LA and Gupta LP, J Vinyl Technol 15:100 (1993). 18 Krigbaum WR and Salaris F, J Polym Sci Polym Phys Ed 16:883 (1978). 19 Krigbaum WR, Lader HJ and Ciferri A, Macromolecules 13:554 (1980). 20 Meneczel L and Wanderlich B, J Polym Sci Polym Chem Ed 18:1433 (1980). 21 Mackley MR, Pinaud F and Siekmann G, Polymer 23:1269 (1982).

Polym Int 51:183±189 (2002)

22 McFarlane FE and Davis TG, US patent 4,001,199 (1977). 23 Farahat MS, Abdel-Azim AA and Abdel-Raowf ME, Macromol Mater Eng 283:1 (2000). 24 Tong SN, Chen DS and Chung LZ, Polymer 24:469 (1983). 25 Rebeiz KS, Fowler DW and Paul DR, J Appl Polym Sci 44:1649 (1992). 26 Abdel-Azim AA and Attia IA, Polym Adv Technol 6:688 (1995). 27 Lee S-C, Sze Y-W and Lin C-C, J Appl Polym Sci 52:869 (1994). 28 Siggia S, Quantitative Organic Analysis via Functional Group Analysis, 3rd edn, John Wiley and Sons, (1963). 29 Stetzler RS and Smullin CF, Anal Chem 34:194 (1962). 30 Fritz JS and Schenk GH, Anal Chem 31:1808 (1959). 31 Fritz DF, Sahil A, Keller H-P and Kovats ECz, Anal Chem 51:7 (1979). 32 Sorenson WR, Sweeny W and Campbell TW, Preparative Methods of Polymer Chemistry, 3rd edn, Wiley-Interscience, New York. p 358 (2001). 33 Odian G, Principles of Polymerization, 2nd edn, John Wiley and Sons (1981). 34 Chiou JS and Paul DR, J Polym Sci Polym Phys Ed 25:699 (1987). 35 Weinkauf DH and Paul DR, J Appl Polym Sci 29:329 (1991). 36 Kang T-K and Ha C-S, J Appl Polym Sci 73:1707 (1999). 37 Li X-G, J Appl Polym Sci 73:2921 (1999). 38 Li X-G and Huang M-R, J Appl Polym Sci 73:2911 (1999). 39 Jackson WJ Jr and Kuhfuss HF, J Polym Sci Polym Chem Ed 14:2043 (1976). 40 Shinn TH, J Appl Polym Sci 49:1093 (1993). 41 Quach L, Hornbogen E, Volksen W and Economy J, J Polym Sci Part B 27:775 (1989). 42 Nicely VA, Dougherty JT and Renfro LW, Macromolecules 20:573 (1987). 43 Zachariades AE and Porter RS, High Modulus Polymers, Marcel Dekker, p 105 (1988). 44 Jackson WJ and Kuhfuss HF, J Polym Sci Polym Chem Ed 14:2043 (1976). 45 Baird DG, Polymeric Liquid Crystals Ed by Blumstein A, Plenum, New York. p 119 (1993). 46 Wissbrun KF, Br Polym J 12:163 (1980). 47 Jerman RE and Baird DG, J Rheol 25:275 (1981). 48 Sugiyama H, Lewis DN, White JL and Fellers JF, J Appl Polym Sci 30:2329 (1985). 49 Gotsis AD and Baird DG, J Rheol 29:539 (1985). 50 Acierno D, LaMantia FP, Polizzotti G, Ciferri A and Valenti B, Macromolecules 15:1455 (1982). 51 Harper CA, Handbook of Plastics, Elastomers and Composites, McGraw-Hill, New York. Chapter 4.3 (1996).

189