Predicting Chain Conformation and Entanglement of Polymers From Chemical Structure

SOUHENG WU*

Science and Engineering Laboratories Experimental Station The D u Pont Company

Wilmington, Delaware 19880-0356

The characteristic ratio C, and the entanglement molecular weight M e are two key molecular parameters that control melt viscoelasticity, solid mechanical (brittle/ductile) behavior, and adhesion of polymers. We show that the character- istic ratio C, and the entanglement molecular weight M e can be predicted from chemical structure by group additivity with uncertainties usually less than - 7% for C, and - 15% for M e , comparable with the accuracies of experimental values.

INTRODUCTION

he characteristic ratio C, and the entanglement T molecular weight M e are two important chain parameters that control melt viscoelasticity (l), solid mechanical (brittle/ductile) behavior ( 2 ) , and adhe- sion (3) of polymers. In this work, we present a group contribution method for predicting the char- acteristic ratio C,, the entanglement molecular weight M e , and the entanglement density v, of poly- mers from chemical structure by group additivity.

CHARACTERISTIC RATIO

The characteristic ratio C, is a measure of the intrinsic flexibility and rigidity of a coiled chain, given by (4, 5)

C, = Lim [ ( R z ) / n ( l : ) ] ( 1 ) n-m

where ( R z ) is the mean-square end-to-end distance of an unperturbed coiled real chain, n the number of statistical skeletal units in the chain, 1, the length of a statistical skeletal unit, and ( 1 : ) the mean-square length of a statistical skeletal unit. A statistical skeletal unit is a real or virtual skeletal bond that is an elementary rotational unit of the main chain ( 2 , 5). further discussed later. Note that our statisti- cal skeletal unit is not a Kuhn step length. Through- out this work, we will use the term “statistical skele- tal unit” to mean “a real or virtual skeletal bond that is an elementary rotational unit.”

The intrinsic viscosity [q0Is of an unperturbed coiled chain is given by (4)

*Present address: Polymer Innovations Consulting. 7 1 1 Taunton Rd.. Wilmington, DE 19803-1709.

where K , is a constant depending on chemical structure, M the molecular weight, $Jo = 2.51 x loz3 ~ m ~ m o l e ’ / ~ g ~ ~ / ~ , and (R;) the mean-square end- to-end distance of an unperturbed chain. Combining Eqs 1 and 2, we obtain

Cm= ( K ~ / $ J , ) ~ ’ ~ [ M ~ / ( ~ : ) ] ( 3 )

where M u is the average molecular weight per statis- tical skeletal unit. Note that M , = M , / n u , where M , is the molecular weight of a repeat unit, and n, the number of statistical skeletal units (real and virtual skeletal bonds) in a repeat unit.

ENTANGLEMENT MOLECULAR WEIGHT

The entanglement molecular weight M e (i.e., the molecular weight of an entanglement strand be- tween two adjacent entanglement junctions in a chain) is given by ( 1 )

M e = pRT/Gg (4)

where p is the melt density, R the gas constant, T the temperature, and G i the plateau modulus. Ex- perimental methods for determining G i have been discussed in detail (5, 6).

On the other hand, we have shown that the entan- glement density increases with increasing chain tor- tuosity (5, 7). The length of an entanglement strand (i.e., between two entanglement points) is given by (5)

Nu = 3C; (5) where Nu is the number of statistical skeletal units in an entanglement strand. Since M , = N U M U , we thus have

M e = 3M,C; (6)

POLYMER ENGINEERING AND SCIENCE, JUNE 1992, Vol. 32, No. 12 823

Combining Eqs 3, 4, and 6, we obtain

S o u h e n g Wu

H , 5.74 ; -- C, = ( pRT/3MuGi) 1'2 (7)

and

ENTANGLEMENT DENSITY

The entanglement density ve is defined as the number of entanglement junctions in a unit volume, i.e.,

ve = pa /Me (9)

where pa is the amorphous density. Combining Eqs 6 and 9, we have

and

VIRTUAL SKELETAL BONDS

A statistical skeletal unit is a real or a virtual skeletal bond, which is an elementary rotational unit on the main chain. The conformation of a chain is determined by the populations of the rotational states of statistical skeletal units. Factors that should be considered in determining a statistical skeletal unit include bond rotational states, coplanarity, and tautomerism of skeletal bonds. For instance, in a saturated vinyl chain, each skeletal single bond (C-C) can rotate around its own axis, and so is a statistical real bond. On the other hand, the C-C bonds in an aromatic ring cannot rotate around their own axes. Therefore, a para-phenylene group with the two connecting single bonds (i.e., -+-) is an elementary rotational unit, and so is a statistical virtual bond.

Figure I illustrates the virtual skeletal bonds in a number of polymers, explained below.

In polyethylene (PE), each skeletal C-C bond is a st2tistical unit with a bond length of 0.153 nm (1.53 A). The same applies to other saturated vinyl polymers, such as polypropylene. polyisobuty- lene, polystyrene, poly(viny1 acetate), poly(viny1 chloride), poly(styrene-stat-acrylonitrile). polyacry- lates, and polymethacrylates.

In polyoxyethylene (POE), the C-C bo2d is a sta- tistical unit of length 0.153 nm (1.53 A), and the C-0 ban$ is another statistical unit of length 0.143 nm (1.43 A).

In polyamides, the amide group is rigid, because of keto-enol tautomerism, which also shortens the C -N bond, i.e.,

-c( =o)-NH-- -c(oH)=N-

Either CH2-C(=O)-NH or CH,-NH-C(=O) may be con2idered as a virtual boond of length 0.246 nm (2.46 A) or 0.242 nm (2.42 A), respectively. There- fore, the segment CH,-C(=O)-NH-CH, is consid-

POE /O

H Polyamide 1

PC

/'$'

Fig. 1 . Virtual skeletal bonds. The bond lengths are in A.

ered t? consist of one virtual bond of average length 2.44 A, and a real bond of average length 0.148 nm (1.48 A).

In poly(ethy1ene terephthalate) (PET), the segment C(=O)-+,C(=O) is a virtual bond of length 0.574 nm (5.74 A). In poly(ethy1ene isophthalate) (PEI), the segment C(=O)-4 i s a virtual bond of length 0.377 nm (3.77 A). Therefore, the segment C(=O)-+-C(=O) consists of one ovirtiial bond C(=O)-+ of length 0.377 nm (3.77 A), andoa real bond C-C(=O) of length 0.147 nm (1.47 A). The same applies to polyarylates, such as the polyary- lates of bisphenol-A with terephthalic acid (PAT) or with isophthalic acid (PAI).

In poly(2.6-dimethyl- 1.4-phenylene oxide) (PPO), the segment O-@CH,)-O is a virtual bond of length 0.562 nm (5.62 A).

In polycarbonate (PC), each of the segments C(=O)-0 and 0-+-CH(CH,) may be considered a rotational unit. In this case, a repeat unit of PC would consist of four real and virtual skeletal bonds. However, we adopt the molecular model of Yannas and Luise (8). which indicates that the segment C(=O)-0-+-C(CH,) should be considered as a ro- tational unit. This virtual skeletal bond has a length of 0.661 nm (6.61 A). In this case, a repeat unit of PC consists of two virtual skeletal bonds. This con- vention is used for all carbonate structures (includ- ing PC and TMPC) throughout this work.

The above examples illustrate how the virtual skeletal bonds may be consistently specified. Note that all chemical bonds are rigid, since they are not

824 POLYMER ENGINEERING AND SCIENCE, JUNE 1992, Vol. 32, No. 12

Predicting Chain Conformation and Entanglement

deformed (bent, stretched, or squeezed) in forming a coil. Rather, it is the rotation of chain segments around the chain axis that gives rise to the chain flexibility and determines the chain conformation. Therefore, it is important to determine the elemen- tary rotational units carefully. To accurately do so, a complete rotational energy spectrum is needed. However, the above examples illustrate how the vir- tual skeletal bonds may be qualitatively but consis- tently identified without using the rotational energy data.

GROUP CONTRIBUTION METHOD

The intrinsic-viscosity constant for an unper- turbed coil chain K , in Eq 2 is a group additive function, given by (9)

K O = ( L r / M r ) ’ (12)

(13) L, = C K , + B n ,

where L , is the molar stiffness function of a repeat unit, K i the molar stiffness constant for group i, B a constant taken to be 4.2 for conventional polymers, n, the number of skeletal atoms in a repeat unit, and the summation C is to be taken over a repeat unit.

The skeletal atoms are the atoms on the main chain, not counting side groups and branches. For instance, all vinyl chains have two skeletal atoms per repeat unit. If an aromatic ring is part of the main chain, n, is counted as 2 for o-phenylene, 3 for m-phenylene, and 4 for p-phenylene (9).

We have previously studied the effects of tacticity on chain conformation and entanglement (10). Us- ing these results in the present work, we find that B varies with the tacticity of main chain, given by

B = 3.2r + 7.1 m

where r is the fraction of racemic (syndiotactic) diads, and m is that of meso (isotactic) diads. For conventional free-radical polymerized poly(methy1 methacrylate) (PMMA), typically we have r = 0.75 and m = 0.25, thus B = 4.12. It is interesting to note that this value is equal to the one originally assigned to conventional polymers by Van Krevelen (9).

Table 1 lists the K i values for a number of groups, compiled from Van Krevelen (9) and the results of our present work. By using these group constants, we can calculate the K O value, and then predict the chain parameters C,, M,, and u, from chemical structure. Using Eqs 12 and 13 in Eqs 3, 8, and 1 I , we obtain the necessary equations as

(14)

Table 1. List of Group Constants.

- CH3 3.55 - CH -CH2 ~ 2.35 - OH > C H - 1.15 -0- > c < 0 - COOH 8.0 - C(:O)O - - CI 8.62

5 8 5.1 (alkylether) 0.1 (arylether) 9.0 6.4 (acrylic)

- oc(=o)o - 27.5 - CONH - 12.6 - so2 - 12.0

16.3 > Si < 5.0

a 18.25

27.5

CH3 \

10.0

8.0

~

parameters from chemical structure by group contribution.

Table 2 lists the mean-square length of a statisti- cal skeletal bond ( l : ) for a wide variety of polymers, calculated by considering the real and virtual skele- tal bonds, as discussed earlier. Figures 2, 3, and 4 show the chemical structures for a number of polymers.

To widen the variety of polymers tested, we have also measured the plateau modulus G i for two polyesters and seven polyarylates by dynamic linear viscoelastic spectroscopy. The experimental method has been discussed in detail before (2, 5, 6). The Me and C, are then calculated from G i by Eqs 4 and 7, respectively. The experimental results are listed in Table 3.

Table 2 compares the experimental and predicted values of C,, M e , and v, for a wide variety of poly- mers. Good agreement between the predicted and the experimental values can be seen.

Regression analysis shows that the standard devi- ation of predicted C, values is - 7%. whereas those for the predicted M, and v, values are both - 15%. These standard deviations are comparable with the uncertainties of experimental values for C,, M e , and v,, and are also consistent with the relationship Nu cc C z , i.e., Eq 5.

Thus, the present group contribution method can adequately predict the chain conformation and en- tanglement behavior for a wide variety of chain structures.

We comment on some molecular-architectural effects below.

GROUP CONSTANT K, The group constant K i is independent of the posi-

tion of the group in the chain, i.e., regardless of

POLYMER ENGINEERING AND SCIENCE, JUNE 1992, Vol. 32, No. 12

Souheng W u

Table 2. Experimental and Predicted Values of C,, Me and ue .

Pa, c, M e , glmol u e . mmolelcc at 25°C

Polymer 25°C A2 Exptl Predict Exptl Predict Exptl Predict glcc (loZ"),

Vinyl polymers PE PIB PVC PVAc Styrene-acrylonitriles SAN25 PAN Polystyrenes PS iPS SPS PaMS Acrylic polymers PMA PMMA iPMMA sPMMA PEMA PnBMA PtBMA PnHMA PEBMA PnOMA PnDMA PnDDMA PnTDMA PnODMA PnDocMA PChEMA PCyHMA PtBCyHMA PPhMA PtBPhMA PChPhMA PbPhMA PDPhMMA PTPhMMA Polyamides N6 N66 APA Silicone PDMS Aliphatic polyethers POM POE Aromatic polyethers PPO PHO Polysulfone PSO Polycarbonates PC TMPC Polyesters PET PEI Polyarylates PA1 PAR90 PAR70 PAR50 PAT TBU TME

0 855 0 840 1 4 0 1 1 9

1 08 1 1 6

1 0 5 1 0 5 1 0 5 1 0 7

1 1 1 1 1 7 1 1 7 1 1 7 1 1 2 1 0 5 1 0 2 1 0 1 1 0 4 0 971 0 937 0 929 0 921 0 904 0 895 1 2 2 6 1 066 0 985 1 1 4 8 1 2 8 7 1 200 1 1 5 0 1 1 2 7 1 1 3 3

1 08 1 07 1 2 6

0 970

1 2 5 1 1 2

1 0 7 1 1 8

1 24

1 20 1 2 1

1 3 3 1 3 3

1 21 1 21 1 21 1 21 1 21 1 2 1 1 2 1

2 34 2 34 2 34 2 34

2 34 2 34

2 34 2 34 2 34 2 34

2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34 2 34

2 92 2 92 4 01

2 69

2 04 2 04

31 6 12 4

34 8

43 7 43 7

7 16 3 77

14 3 14 8 16 2 17 4 20 5 14 3 14 3

5 7 6 6 7 6 9 4

10 1 6 3

10 8 13 9

10 1

8 3 7 9

10 7 7 6 8 0 8 3

12 0 11 2

9 5 12 6 13 4 14 5 14 9 20 6 24 1 10 5 11 6 12 0 13 1 14 6 14 1 16 4 14 0 20 3

6 2 6 1 5 4

6 8

7 5 6 9

3 2 3 7

2 2

2 4 3 1

4 10 7 75

3 72 3 65 3 26 3 14 2 55 4 02 4 46

-

5 5 6 5 7 6 8 6

9 0 6 2

10 7 13 3 9 8

11 4

7 4 8 1

10 6 7 2 8 8

10 1 10 3 11 5 11 5 12 9 14 3 15 6 16 4 19 8 22 6 11 2 10 8 11 8 13 0 15 4 15 4 18 2 19 9 24 8

6 0 6 0 6 7

6 3

7 9 6 6

2 9 2 9

2 2

2 7 3 1

3 56 7 75

3 75 3 62 3 33 3 09 2 64 3 89 4 24

1390 591 0 541 0

11,400

11,500 31 20

10,700 20,800

23,000

9070 9200

16,400 7810 8590

12,200 28,100 33,800 22,000 47,100 60,900 80,100 89,200

21 5,000 343,000

24,600 33,900 48,400 44,300 69,700 55,100

96,OO 74,100

203,000

2490 2000 2040

6650

2540 2200

3620 1900

2250

1780 3620

1450 4950

2480 2420 2000 1920 1390 3340 41 20

-

1250 3600 541 0 961 0

10,200 3060

17,900 27,800 14,900 22,900

71 20 9880

16,900 7860

13,200 21,900 22,400 33,800 33,800 49,400 69,100 93,400

107,000 199,000 301,000

27,800 29,500 47,000 41,100 77,600 65,400

1 19,000 99,900

303,000

2040 2040 3390

4380

2790 1940

2950 1220

1610

2550 4440

1210 4950

2520 2380 2080 1870 1500 31 30 371 0

0 615 0 154 0 259 0 104

0 094 0 372

0 056 0 036

0 046

0 122 0 127 0 071 0 150 0 130 0 086 0 036 0 030 0 047 0 021 0 016 0 012 0 010 0 004 0 003 0 050 0 031 0 020 0 026 0 010 0 022 0 012 0 015 0 006

0 434 0 535 0 618

0 146

0 492 0 509

0 296 0 621

0 551

0 674 0 334

0 917 0 269

0 488 0 500 0 605 0 630 0 870 0 362 0 294

-

0 683 0 233 0 259 0 124

0 106 0 379

0 060 0 038 0 070 0 047

0 156 0 118 0 069 0 149 0 085 0 048 0 045 0 030 0 031 0 020 0 014 0 010 0 009 0 004 0 003 0 044 0 036 0 021 0 028 0 017 0 018 0 010 0 01 1 0 004

0 529 0 524 0 372

0 221

0 448 0 576

0 363 0 967

0 770

0 471 0 272

1 099 0 269

0 480 0 508 0 582 0 647 0 807 0 387 0 326

*The standard deviations are - 7 % for the predicted C, values, and - 15% for the predicted Me and v g values, comparable with the uncertainties of experimental values. *All experimental data are taken from Wu (2 , 5), and Tables 3 and 4. The C, values for polymethacrylates are taken from M. Kurata and Y. Tsunashima, in Polymer Handbook, 3rd ed., J. Brandrup and E. H. Immergut, eds., Wiley, New York (1989), VII-1 - 60; J. W. Mays and N. Hadjichristidis, J. Macromol. Sci.-Rev. Macromol. Chem. Phys,, C28, 371 (1988); J. W. Mays, N. Hadjichristidis, and J. S. Lindner, J. Pofym. Sci., B. Pofym. Phys., 28, 1881 (1990); L. Gargallo, N. Hamidi, and D. Radic, Polymer, 31, 924 (1990).

826 POLYMER ENGINEERING AND SCIENCE, JUNE 1992, Vol. 32, No. 12

Predicting Cha in Conformation and Entanglement

whether the group is part of the main chain (skele- ton) or the side chain. However, each skeletal atom

y 3 -CHz-C - -CH - C - - C H 1 C - -CH - C -

I 2 1 I 2 1

PCyHMA

CH 3

- W - C - -CH - C - - C H - C - -CH-C- I 2 1 2 / 2 1

PPhMA PlBPhMA $ H C CH3

CH3

Fig. 2. Chemical structures of polymethacrylates.

CH OH

Jg-$$P o-c-o-

CH 3 'CH /

C" 3

TMPC

0 PSO

Fig. 3. Chemical structures of phenoxy resin [PHO), tetramethyl polycarbonate [TMPC), a n d polysulfone (PSO) .

contributes an additional amount of B to the molar stiffness function, as given by E q 14.

Table 2 shows that the group constants can be used to predict the chain conformation and entan- glement behavior within usual experimental uncer- tainties for wide varieties of chain structures, as already noted.

EFFECT OF TACTICITY

The tacticity of the main chain has a profound effect on chain conformation and entanglement (5, 10). The B value varies with the tacticity, as given by Eq 14. Table 2 shows that the predicted values of C,, M e , and v, agree well with the experi- mental values for conventional PMMA, isotactic PMMA, syndiotactic PMMA, conventional PS and isotactic PS.

8 TY Pe

A

B

C

Fig. 4. Chemical structures ofpolyarylates.

Table 3. Experimental Values of G: and Me, and C, of Polyarylates, Polyesters, and Poly(t-butyl methacrylate) Determined in This Work.

Mw, i / t GN" P at T , Me 9

Name ( i l t ) glmole M,/M, Ratioa T,, "C MQa T , "C glcc glmol C,

Polyarylates

PAl( 1 00 / 0) PAR90(90 I 1 0) PAR70(70 130) PAR50(50/50 PAT( 0 / 1 00)

TBU(10010)

TME( 1 OO/O) Polyesters PET(O/ 1 00) PEI(10010) Polymethacrylate PtBMA

54,900 53,000 60,200 58,600

47,200

47,100

77,600 11 5,000

106.000

2.76 2.29 2.27 2.28

2.42

2.74

2.06 1.83

1.03

10010 901 10 70130 50/50

01100

1 0010

1 0010

01 100 10010

Type A 184 183 199 197 21 6

209

21 3

75 58

120

Type B

Type C

1.98 250 1.122 2480 3.72 1.99 240 1.128 2420 3.65 2.47 260 1.116 2000 3.26 2.57 260 1.116 1920 3.1 4 3.56 260 1.116 1390 2.55

1.50 270 1.109 3340 4.02

1.15 230 1.134 41 20 4.46

3.10 275 0.989 1450 3.89 0.915 150 1.287 4950 7.75

0.106 140 0.982 31,800 12.2 ~~~ ~

' i l t Ratio = isophthalate to terephthalate ratio.

POLYMER ENGINEERING AND SCIENCE, JUNE 1992, Vol. 32, No. 12 827

Souheng Wu

Table 4 compares the predicted and the experi- mental values of C, and M, as a function of tacticity for PMMAs in more detail. Figure 5 plots C, and Log M, vs. the meso (isotactic) diads frac- tion for PMMA. The symbols are experimental. The lines (slightly curved) are predicted by group contri- bution. The standard deviations between the pre- dicted and the experimental values are only 2% for C, and 3% for M e .

SIZE AND SHAPE OF SUBSTITUENT GROUP

In Table 2, the homologous polymethacrylate se- ries shows that the size and shape of substituent groups on the side chains do not require special treatment, and are well accounted for by the sum of the group constants.

For instance, the side chain varies from methyl

Table 4. Experimental and Predicted Me and C, Values as a Function of Tacticity for Poly(Methy1 Methacrylate).

Diads Me, glmole c, m r Predict Exptl Predict Exptl

0 1 7900 0.14 0.86 8900 0.23 0.77 9700 0.33 0.67 10,500 0.37 0.63 10,900 0.50 0.50 12,100 0.81 0.19 15,300 1 0 17,500

7900 91 00 9700 9600

10,100 13,000 14,200 18,000

7.2 7.3 7.7 7.8 8.0 8.0 8.4 8.0 8.5 8.2 9.0 9.3

10.1 9.7 10.8 10.7

3.5 ' I 1 I I I I I I I

15

10 (j

5 0 0 2 0 4 0 6 0 8 1 0

MESO (ISOTACTIC) DIADS FRACTION, rn

Fig. 5. C, and Log Me us. tacticity for PMMA. Symbols are experimental. Solid lines (slightly curved) are pre- dicted by group contributions.

to docosyl ((CH,),,CH 3] , from n-butyl to t-butyl, from cyclohexyl to phenyl, from chloroethyl to chlorophenyl, from t-butylcyclohexyl to t- butylphenyl, from biphenyl to triphenylmethyl. The predicted and experimental values of C,, M,, and v, are within experimental uncertainties.

POSITION OF SUBSTITUENT GROUP

As mentioned before, the group constant K , is independent of the position of the group. However, when several substituent groups are crowded around the main chain to cause unusually large steric hin- drance, the molar stiffness of such a sterically hin- dered structure may be greater than the sum of the K , values of the constituent groups. In such a case, a particular K, value is assigned to the particular sterically hindered structure as a unique structural unit. An example is the 2,6-dimethyl- 1,4-phenylene group ( K , = 27.5 cgs), shown in Table 1 . This K , value is valid for any polymer containing this group, such as PPO and TME, as can be seen in Tables 2 and 3.

It is instructive to compare the behavior of the three polyarylates, PAI, TBU, and TME, listed in Tables 2 and 3. All three have the same skeletal structure, consisting of bisphenol-A and isophtha- late, shown in Figure 4. In TBU, the t-butyl group is attached to the 5-position of the isophthalate group, and so provides little hindrance to the rotation around the isophthalate virtual skeletal bond. Therefore, the behavior of TBU is adequately given by summing the group constants for the consti- tuent groups, i.e., bisphenol-A, isophthalate, and t-butyl groups.

On the other hand, in TME, two methyl groups are attached to the 2,6-positions of phenylene unit, crowding around the skeletal oxygen, and thus con- siderably hindering the rotation around the skeletal bond. Therefore, the 2.6-dimethyl- 1,4-phenylene group has a K , value of 27.5 cgs, which is 6.5 cgs larger than the sum for the constituent groups and atoms, discussed below.

Note that PPO contains the same 2.6-dimethyl- 1,4-phenylene group as TME. The same K , value of 27.5 cgs for this group should be valid, as is indeed found to be true.

In contrast, the sum of the K , values of the con- stituent groups in the 2,6-dimethyl- 1,4-phenylene group gives a K , value of 21.0 cgs, i.e., the sum of one phenylene group ( K , = 16.3 cgs), two methyl groups ( K , = 2 x 3.55 cgs) and minus two hydrogens ( K , = 2 x 1.2 cgs). The difference of 6.5 cgs between the actual K , value (27.5 cgs) and the summed constituent K , value (21.0 cgs) accounts for the increased steric hindrance due to the crowding effect of methyl groups.

The experimental and predicted chain conforma- tion and entanglement behavior of PAI, TBU, and TME are listed in Tables 2 and 3. As the steric hindrance increases, the chains become more ex- panded (i.e., greater C, value) and less entangled

a28 POLYMER ENGINEERING AND SCIENCE, JUNE 1992, Yol. 32, No. 12

Predicting C h a i n Conformat ion a n d En tang lemen t

(i.e., greater M , value). These are correctly pre- dicted by the group contribution method.

Therefore, the value of the individual group con- stant K , is not affected by the size, shape, or posi- tion of the substituent group. An exception occurs when the substituent groups are attached directly to the main chain in such a way that they cause a significantly large steric hindrance to the skeletal bond rotation, such as in the 2,6-dimethyl-l,4-phen- ylene group. In this case, a particular group con- stant value is assigned to the particular group, rather than summing the individual constituent groups.

TEREPHTHALATE VS. ISOPHTHALATE

The effect of steric hindrance is further mani- fested in terephthalate and isophthalate groups on the main chain. The experimental values of plateau modulus and entanglement molecular weight for poly(ethy1ene terephthalate) (PET) and poly(ethy1- ene isophthalate) (PEI) are listed in Table 3. It can be seen that PET is about three times more entan- gled (GE = 3.1 MPa, M e = 1450) than PEI (GE = 0.915 MPa, M e = 4950). This is consistent with their chem- ical structures, which indicate that the isophthalate group is more sterically hindered than the tereph- thalate group, shown in Figure 1. The difference in steric hindrance is reflected in their K , values, i.e., K , = 16.3 cgs for terephthalate group and K i = 40.8 cgs for isophthalate group, given in Table 1.

It is significant that these K i values are also appli- cable in the various polyarylates containing various ratios of terephthalate and isophthalate groups on the main chain, as can be seen in Table 2.

STATISTICAL AND ALTERNATING COPOLYMERS

The present method is equally applicable to statistical copolymers, as we have shown for poly(styrene-acrylonitrile) (SAN), various polyary- lates of different isophthalate/terephthalate ratios, amorphous polyamide (APA) listed in Table 2, and poly(methy1 methacrylate-styrene) and poly(methy1 methacrylate-N-phenylmaleimide) discussed elsewhere (6).

Furthermore, all homopolymers may be regarded as copolymers of their constituent units. For instance, poly(methy1 methacrylate) may be regarded as an alternating copolymer of -CH2- and -C(CH,)C( =O)OCH,--. Therefore, the present method is also applicable to alternating copolymers as well as statistical copolymers.

SUMMARY AND CONCLUSION

We have shown that the chain parameters C,, M,, and u, can be predicted from chemical structure by group additivity usually with standard deviations less then - 7% for C, and - 15% for M , and u,, comparable with the uncertainties of experimental values. The present method has its origin in the pseudotopological theory of chain entanglement (5), and explicitly accounts for the roles of molecular

architecture, intrinsic chain flexibility, chemical composition, and skeletal tacticity.

APPENDIX: LIST OF ACRONYMS

Vinyl Polymers

PE: Polyethylene, see Fig. 1. PIB: Polyisobutylene. PVC: Poly(viny1 chloride). PVAc: Poly(viny1 acetate).

Styrene-acrylonitrile Polymers

SAN25: Poly(styrene-acrylonitrile) with 25 w% of

PAN: Polyacrylonitrile. acrylonitrile (AN).

Polystyrenes

PS: Polystyrene (conventional). ips: Isotactic polystyrene, 100% isotactic (see Refs.

sPS: Syndiotactic polystyrene, 100% syndiotactic

PaMS: Poly( a-methylstyrene).

Acrylic Polymers

PMA: Poly(methy1 acrylate). PMMA: Poly(methy1 methacrylate), conventional,

isotactic diads = 0.25 and syndiotactic diads = 0.75.

iPMMA: Isotactic poly(methy1 methacrylate), 100% isotactic (see Refs. 5, 10).

sPMMA: Syndiotactic poly(methy1 methacrylate), 100% syndiotactic (see Refs. 5, 10).

PEMA: Poly(ethy1 methacrylate). PnBMA: Poly(n-butyl methacrylate). PtBMA: Poly(t-butyl methacrylate). New experimen-

tal data given in Table 3 supercede the previous data in Ref. 5.

5, 10).

(see Refs. 5, 10).

PnHMA: Poly(n-hexyl methacrylate). PEBMA: Poly(2-ethylbutyl methacrylate). PnOMA: Poly(n-octyl methacrylate). PnDMA: Poly(n-decyl methacrylate),

PnDDMA: Poly(n-dodecyl methacrylate),

PnTDMA: Poly(n-tridecyl methacrylate),

PnODMA: Poly(N-octadecyl methacrylate),

PnDocMA: Poly(n-docosyl methacrylate),

PChEMA: Poly(2-chloroethyl methacrylate),

PCyHMA: Poly(cyclohexy1 methacrylate), see Fig. 2. PtBCyHMA: Poly(4-t-butylcyclohexyl methacrylate),

PPhMA: Poly(pheny1 methacrylate), see Fig. 2. PtBPhMA: Poly(4-t-butylphenyl methacrylate), see

-[CH,-C(CH ,)(O=C--O-(CH,),CH3)] -.

- [CHz-C(CH 3)(O=C-O-(CH 2 ) 1 lCH,)]-.

-[CH,-C(CH ,)(O=C-O-(CH 2 ) l2CH 3)] -.

--(CH,C(CH ,)(O=C-O-(CH 2 ) 17CH3)] -.

- [CH,C(CH ,)(O=C-O-(CH 2)21CH 3)] -.

--[CH2C(CH3)(O=C-O-CH,CH2Cl)]-.

see Fig. 2.

Fig. 2.

POLYMER ENGINEERING AND SCIENCE, JUNE 1992, Vol. 32, No. 12 829

S o u h e n g Wu

PChPhMA: Poly(4-chlorophenyl methacrylate), see

PbPhMA: Poly(4-biphenyl methacrylate), see Fig. 2. PDPhMMA: Poly(diphenylmethy1 methacrylate), see

PTPhMMA: Poly(triphenylmethy1 methacrylate), see

Fig. 2.

Fig. 2.

Fig. 2.

Polyamides and Polyimide

N6: Polycaprolactam (nylon 6). N66 : Poly(hexamethy1ene adipamide) (nylon 66). APA: Amorphous polyamide, condensation product

of hexamethylenediamine, isophthalic acid (70%), and terephthalic acid (30%).

Silicone

PDMS: Polydimethylsiloxane.

Aliphatic Polyethers

POM: Polyoxymethylene. POE: Polyoxyethylene, see Fig. 1 .

Aromatic Polyethers

PPO: Poly(2.6-dimethyl- 1,4-phenylene oxide), see

PHO: Phenoxy resin, i.e., poly(bispheno1-A-1.3-(2- Fig. 1.

hydroxyl)propane), see Fig. 3.

Polysulfone

PSO: Polysulfone of bisphenol-A and diphenylene- sulfone, see Fig. 3.

Polycarbonate

PC: Polycarbonate of bisphenol-A, see Fig. 1 . TMPC: Poly(3,3’,5,5’-tetramethyl bisphenol-A

carbonate), see Fig. 3.

Polyesters

PET: Poly(ethy1ene terephthalate), see Fig. 1. PEI: Poly(ethy1ene isophthalate), see Fig. 1.

Polyarylates

PAI: Polyarylate of bisphenol-A and isophthalic acid, Type A in Fig. 4.

PARSO: Polyarylate of bisphenol-A, isophthalic acid (90%) and terephthalate (10%). Type A in Fig. 4.

PAR70: Polyarylate of bisphenol-A, isophthalic acid (70%), and terephthalic acid (30%). Type A in Fig. 4.

PAR50: Polyarylate of bisphenol-A, isopht halic acid (SO%), and terephthalic acid (50%), Type A in Fig. 4.

PAT: Polyarylate of bisphenol-A and terephthalic acid, Type A in Fig. 4.

TBU: Polyarylate of bisphenol-A and 5-t-butyl isoph- thalate, Type B in Fig. 4.

TME: Polyarylate of 2,2’,6,6’-tetramethyl bisphenol- A and isophthalate, Type C in Fig. 4.

REFERENCES

1. J. D. Ferry, Viscoelastic Properties of Polymers, 3rd

2. S . Wu, Polym. Eng. Sci., 30, 753 (1990). 3. S. Wu, Polymer Interface and Adhesion, Marcel

4. P. J. Flory, Statistical Mechanics of Chain Molecules,

5. S. Wu, J. Polym. Sci., B. Polym. Phys., 27, 723 (1989). 6. S. Wu and R. Beckerbauer,, to be published in Poly-

7. A. E. Tonelli, J . Polym. Sci. A-2, 8, 625 (1970). 8. I. V. Yannas and R. R. Luise, Macromol. Sci. Phys.,

9. D. W. van Krevelen, Properties of Polymers, Elsevier.

ed., Wiley, New York (1980).

Dekker, New York (1982).

Interscience, New York (1969).

mer.

B21, 443 (1982).

Amsterdam (1976). 10. S. Wu and R. Beckerbauer, to be published.

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