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THE COLLEGE OF AERONAUTICS

CRANFIELD

THE TEMPERATURE DEPENDENCE OF PHOTO-ELASTIC

PROPERTIES OF CROSS-LINKED AMORPHOUS POLYETHYLENES

by

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CoA Note Mat. No. 11

August^_1967

DEPARTMEMD OF MATERIALS

The temperature dependence of photo-elastic properties of

cross-linked amorphous polyethylenes

by

-D.W. Savinders, D.R. Lightfoot eind D.A. Parsons

S U M M A R Y

Cross-linked samples of polyethylene were prepared by e l e c t r o n

i r r a d i a t i o n of both high and low density polymers i n the c r y s t a l l i n e s t a t e .

A further cross-linlted sample was obtained by curing a high density polyethylene

by r e a c t i o n with dicumyl peroxide a t l80°C. The s t r e s s - s t r a i n birefringence

r e l a t i o n s were obtained, on specimens cut from these samples, a t temperatures

between I50 and 250°C.

All samples showed a s u b s t a n t i a l decrease in s t r e s s - o p t i c a l coefficient

with increasing degree of c r o s s - l i n k i n g and with increasing temperature. The

s t r e s s - o p t i c a l p r o p e r t i e s at each temperature were extrapolated to zero degree

of c r o s s - l i n k i n g to give q u a n t i t i e s c h a r a c t e r i s t i c of the Guassian network.

Comparison of these p r o p e r t i e s with Gaussian theory of the network leads to a

value of ca.1150 cals/mole for the difference i n energy between t r a n s and

p;auche confoimations of successive l i n k s of the polyethylene chain and also

i n d i c a t e s t h a t the o p t i c a l anisotropy of a - CH2 - group in the elastomeric

s t a t e i s more nearly given by Denbigh' s than by Bunn and Daubeny' s p o l a r i s

-a b i l i t i e s .

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Page No.

Sviramary

Introduction 1

Theoretical background 1

Experimental

h

Determination of the stress-strain birefringence

relations 5

Determination of densities and refractive indices 6

Results 6

Dependence of stress-optical properties on degree

of cross-linking 7

Dependence of s t r e s s - o p t i c a l p r o p e r t i e s on temperature 8

Conclusions 11

References 11

Tables 15

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1

-Introduction

The well-known k i n e t i c theory of r u b b e r - l i k e e l a s t i c i t y may be extended

t o describe the p h o t o - e l a s t i c p r o p e r t i e s of cross-linl^ed networks, i f the

v a l e n c e - o p t i c a l scheme involving the p r i n c i p l e of t e n s o r i a l addivity of

o p t i c a l p o l a r i s a b i l l t i e s i s accepted. The p h o t o - e l a s t i c theory and t h i s

v a l e n c e - o p t i c a l scheme have been discussed in some d e t a i l by Treloar (1958)

and Volkenstein (1965).

By making use of the r o t a t i o n a l - i s o m e r i c model of a polyethylene chain,

Saunders (l957) derived a t e n t a t i v e value for the difference in energy, U,

between the t r a n s and gauche conformations of successive l i n k s of the chain

from p h o t o - e l a s t i c measurements on cross-linked polyethylene a t a single

temperature above the crystal-melting p o i n t . The derivation involved the

assumption of a p a r t i c u l a r value f o r the o p t i c a l anisotropy of the monomeric

u n i t of the chain and the a c t u a l valvie chosen i s ncni open to some c r i t i c i s m .

The value obtained for U was 2,500 cals/mole, which sviggested the polyethylene

chain was considerably ' s t i f f e r ' than might have been expected from estimates

of U for short chain p a r a f f i n .

C i f e r r i , Hoeve and Flory (1961) obtained an independent estimate of

U = 500 cals/mole from s t r e s s - t e m p e r a t u r e measurements on cross-linked

polythene above i t s melting p o i n t .

I n t h i s paper, r e s u l t s of measurement of the temperature dependence of

the p h o t o - e l a s t i c p r o p e r t i e s of cross-linked polyethylene above t h e i r c r y s t a l

melting point are presented. I t i s shown t h a t these can be described by the

theory if a value of U = II50 cals/mole i s adopted. This r e s u l t does not

depend on the assumption of a p a r t i c u l a r value for the o p t i c a l anisotropy of

a monomeric u n i t , indeed i t i s shown that an independent estimate of t h i s

quantity can be obtained from the measurements.

Theoretical background

I n i t s c l a s s i c a l form the k i n e t i c theory of rubber-lilte e l a s t i c i t y (see

T r e l o a r , 1958) was developed for a network of long chains of randomly-jointed

l i n k s . The r e s u l t i n g load-deformation r e l a t i o n i n simple tension may be

expressed in terms of the t e n s i l e s t r e s s t and the extension r a t i o X as

i n which f i s the applied t e n s i l e force acting on a c r o s s - s e c t i o n a l area of

magnitude AQ when undeformed. N i s the number of a c t i v e network chains per

u n i t volume of the network, k i s Bolt2marjn's constant, T is the absolute

temperature, P i s the density of the cross-linlied network, R the gas constant

and M the number aversige network chain molecular weight.

I f each l i n k i s assumed to be o p t i c a l l y a n i s o t r o p i c with \iiiiaxial

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symmetry about tlie link direction, so that its properties may be characterised by optical polarisabillties q^ and a^ respectively for light polarised with electric vector parallel and perpendicular to the link direction, then the

birefringence of the stretched network An, may be computed from the distribution of link directions predicted by the theory by ass\jming the usual tensorial

additivity of bond polarisability as discussed elsevrhere, see for example Volkenstein (1965). This leads to

An = n, . n3 = i £ f ^ I f (q, - a j (x= . i ) (.)

in which ni and na are respectively the refractive indices for light polarised with electric vector parallel and perpendicular to the direction of extension and iï is the mean refractive index of the network, which in a defoimation at constant volume is taken to be independent of deformation.

The stress optical coefficient is thus given by

This is independent of degree of cross-linking except insofar as cross-linking results in volume changes or changes in chemical composition. Since these are usually talken to be small the stress-optical coefficient is usually taken to be sensibly independent of degree of cross-linking.

The behaviour of real molecular networks may be more nearly approximated by considering networks of chains in which there is some correlation in

direction of successive links due to valence angle restrictions and variations of internal energy during rotation about single bonds. A simple form of

such a chain may be taken to consist of n^ rigid identical linies each of length Z^ (Saunders, 1957)(Volkenstein, I965) when, within the limitations of nj^

large and the end-to-end vector rj^^ small so that i-jjj « n^j^jj^, the distribution of r_ vectors for the free diain may be taken to be Gaussian and we may write the folloa^ing relations for the mean square length Ï ^ , the fully extended length R^, the optical anisotropics by^ (difference between principal

polar-isabillties for the whole free chain, which can be shown to have uniaxial optical symmetry about the r^j^ direction) at lengtli r.^, and bT^ at length R^^.

7 ^

=

Kfn

l^

(a)

m -^ m m ^ R = Kan-E' (b) m ^ m m ' j, (i^^ r„2

W .

K,n

(ftj, - 3 )

(d)

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5

-in which (P.{,-P ) i s the difference -in p r i n c i p a l o p t i c a l p o l a r i s a b i l l t i e s

for a single l i n k , taken to be p a r a l l e l and perpendicular to the l i n k

d i r e c t i o n respectively and Ki, KQ, K3 and K4 are parameters describing

the l i n k - t o - l i n k c o r r e l a t i o n in d i r e c t i o n . In general these w i l l be

functions of valance angle, the form of the p o t e n t i a l energy d i s t r i b u t i o n

for r o t a t i o n about each l i n k and temperature. When K^ = K2 = K3 = K4 = 1

the expressions reduce to those for a chain of randomly-jointed l i n k s as

considered in the c l a s s i c a l theory.

I t has been shown, Volkenstein and P t i t s y n (l955)> F l o r y . Hoeve and

C i f e r r i (l959)> t h a t for a network of such chains equation ( l ) above is

replaced by

in which r ? i s the mean square end-to-end distance for the N active chains

per u n i t volume of the network in the i s o t r o p i c , f o r c e - f r e e , condition and

f2 i s as given in equation (li-a) for the free d i a i n . This modification t o

equation ( l ) makes an allowance for the intramolecular change in i n t e r n a l

energy in the sample during deformation and also obviates the assumptions,

made in the c l a s s i c a l form of the theory, concerning the d i s t r i b u t i o n of

end-to-end vector_lengths in the undeformed network. I t should be noted

/r?\

t h a t the factor I •== I i s temperature dependent; r": can be taken t o depend on

the thermal e^qpansion of the i s o t r o p i c network whereas r^ w i l l have a

temperature dependence determined by the form of Kf for the p a r t i c u l a r type

of chain considered.

Following the method for c a l c u l a t i n g the birefringence of the network

e s t a b l i s h e d by Treloar (19^7)^ but using equations (5) r a t h e r than those for

randomly-joined chains, and f u r t h e r not making any assumptions concerning

the value of the mean square length of the end-to-end vectors in the

undeformed network, i t i s easy to show t h a t the birefringence of the

deformed network may be w r i t t e n

^ = i ^ . | S K 3 K f ( 2 ) 0 ^ ^ „ K x 3 . ^ ) (6)

m

From (5) and (6) the s t r e s s - o p t i c a l coefficient therefore becomes

which i s i d e n t i c a l with the expression given by Saunders (1957) in which the

t "" 5

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importance of this factor has been stressed by Flory et al (l959); its

inclusion in equation (5) represents the most rigorous method yet devised

of allowing for internal energy effects in calculating the free energy

of deformation of the network. It will be noted that it modifies the

tonperature dependence of stress in the deformed network from the near

proportionality to absolute temperattire suggested by equation (l).

Equation (7) shows that by considering chains involving energetic effects

the temperature dependence of the stress optical properties of the deformed

network is also modified.

It is convenient to define a quantity A by the relation

A _ A n 5 U^kT /ON

^

"

t (H2+2)2

2K ^^

which includes only universal constants and properties of the network

which are directly available to measurement. We may now re-write equation

(7) in the form

A = K 3 K ! 0 ^ - P ^ ) (9)

in which we note that the right hand side contains only quantities describing

the properties of a free chain. We can thus infer properties of the free

chain from suitable measurenEnts on the network. We ftirther note that since

(^'t'^m^ is essentially independent of temperature, the observable temperature

dependence of A is given by the temperature dependence of K3K1, a combination

of parameters describing the link-to-link correlation in the free chain.

Information obtained from a study of the temperature dependence of

stress-optical properties should therefore be complementary to information obtained by

Flory, Hoeve and Ciferri (1959), Ciferri, Hoeve and Floiy (1961) from

experiments on the temperature dependence of stress in deformed networks which

gives information on Kf via equations 4(a) and 5.

Experimental

Materials: Experiments were carried out on samples of three different tjrpes

polyethylene. These were l) a low density polyethylene designated DYNK

obtained from Baltelite Limited, 2) a high density polyethylene designated

HIFAX 1600 obtained from the Hercules Powder Co. Ltd., and 5) a high density

polyethylene Hostalen G.S. obtained from Hoechst Chemicals Ltd.

The DYNK Al© HIFAX polymers were pressed into sheets 1 to 2 mm thick,

annealed, and subsequently cross-linked by exposure to electrons from a

2 m.e.V. linear accelerator. Irradiation was carried out at room temperatui«

and samples were subjected to nominal doses of 20, lj-0, 80 and I60 m.rad.

The results reported refer, therefore, to polymers free from such additives

as antioxidants.

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5

-The Hostalen samples were prepared from polymer obtained in powder form. Up to ten parts by weight dicumyl peroxide per h\mdred parts of polymer was incoorporated in the po\^ered polymer by a solution technique and cross-linked sheets between 1 and 2 mm thick were prepared by pressing for 8 mins. at l80''C. The sheets were subsequently annealed by heating in nitrogen.

Samples were prepared for the experiments in the fozm of dumbells cut from the cross-linked sheets.

Determination of the stress-strain birefringence relations

The stress-strain-birefringence relations were determined for each sample at various deformations during one complete cycle of loading and unloading at constant known temperatvires in the range 150°C to 240''C, i.e. at temperatures above the crystal melting point for the cross-linked materials. Simple tensile deformations were used throughout and the

extension ratios were restricted to values less than those likely to induce crystallisation. The extension ratios rarely exceeded a value X = 2.

The specimens were enclosed in a small fxirnace in which the temperature could be controlled to better than ± I'C. The furnace was kept filled with oxygen-free nitrogen to minimise oxidative degradation. Extension ratios were derived from the separations of two ink marks on the specimen surface as

determined using a cathetometer. The tensile loads were applied via a

calibrated spring system. The birefringence was determined from measurement of optical path difference in the specimen using either a Babinet or

Senarmont compensator.

In order to calculate the stress, strain abd birefringence it was necessary to know the relevant dimensions of the samples in the unstrained state at

the temperature of each experiment. The distance between the marks in the unstrained state was determined, in each case, by a suitable extrapolation of load-length measurements at the required temperature. The widths and the thicloiesses of the specimens at each temperatu.re were calculated from the widths and thiclcnesses measured at room tanperature and the measured overall

expansion of each sample between room temperature and the required temperature. This latter quantity was determined by observing the separation of two marks

on a sample as a function of temperature using a cathetometer. In subsequent calculations it was assumed that the samples were isotropic, this assumption was substantially borne out by the results of subsidiary experiments on samples bearing marles in various orientations.

Samples were pre-conditioned by taking them throiigh a complete cycle of loadiiog and unloading at the required temperature prior to malcing measurements. During measurements the following sequence was used (i) deformation applied

(ii) separation of marks observed (iii) path difference observed (iv) load observed. A rigorous time pattern of measurement was not imposed throughout but in each case sufficient time was allowed for the measui-^d path difference and loads to have achieved substantially constant values.

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Stresses and birefringence were calculated from the measured quantities on the assumption that the volume of the sample remained constant during deformation at constant temperatLure.

Determination of densities and refractive indices

It is necessary to know the density and the refractive index of each sample in the undeformed state at each temperature. The densities were obtained from the densities determined at room temperature, by hydrostatic weighing and flotations measurements, and the linear e:cpansions determined

as described above.

Refractive indices were calculated from the densities using the value of 0.5285 cc/gm for the specific refractivity of polyethylenes as given by Bianchi, Luetzel and Price (1958).

Results

I n a l l cases the r e l a t i o n between the b i r e f r i n g e n c e . An, and the t e n s i l e s t r e s s , t , was l i n e a r and r e v e r s i b l e within the accuracy of the measurements. The neasurements for the l o a d - i n c r e a s i n g p a r t of the cycle were on 'the same l i n e as those f o r the load decreasing p a r t of tlie c y c l e . G?he slope of the l i n e was taken as the s t r e s s - o p t i c a l c o e f f i c i e n t and designated ^ .

The r e l a t i o n between the s t r e s s , t , and e x t e n s i o n - r a t i o , X^ was such t h a t p l o t s of

t vs (x- . i )

were substantially linear and reversible. There were some systematic

departures from linearity and reversibility, particularly in the most lightlj'' cross-linked samples, but these were relatively small. It should be noted that invariably the amplitude of the imposed strain cycle increased as the degree of cross-linking decreased and vice versa for the amplitude of the stress cycle. No investigation of the departures from reversibility and linearity will therefore be undertaken. Values of

X X=l

were taken from the graphs .

The values cf A and rr- were c a l c u l a t e d from the measured q u a n t i t i e s c

according to equations (8) and ( l ) r e s p e c t i v e l y . The significance of A lias already been explained. The values of i - w i l l be used as a simple comparative

Mc

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7

-obtain a more sophisticated value of — by a more detailed analysis such c

as has been carried out for natiiral rubber, see Mullins and Thomas (1965) • and attempted for polyethylene by Vickroy and Gent (1966).

The full resvilts are given in Tables 1, 2 and

5-There are two sets of results for Hifax 1600 given in parts (a) and (b) of the table respectively. These results all relate to samples of tlie same polymer but the conditions of irradiation varied between parts (a) and (b) and the nominal doses are therefore not comparable. This is borne out in figiure 5 in which it will be seen that the SOffi sample of part (b) corresponds more closely with the k(MR sample than with SQ-IR sample of

part (a).

Dependence of stress-optical properties on degree of cross-linleing

Inspection of Tables 1, 2 and 3 shows that for all three sets of

samples the stress-optical coefficient, and hence the value of A, decreases as the degree of cross-linlving increases. This is shown more clearly in

Figures 1, 2 and 3 in which the values of A are plotted against ^ . Despite

Mc

1

a pronoimced scatter it is evident that A decreases substantially as — increases for all samples. This resvilt is in agreement with the earlier work of Saunders (1956, 1957)^ in which such an effect was first noted. It was suggested there that such behaviour could be attributed to short-chain non-Gaussian behaviour in the network, but quantitative agreement with non-Gaussian theory could onlj'' be obtained if the polyethylene chain was taken to be relatively stiff compared with, say, the rubber and gutta percha chains, so that 'short-chain' effects became more important.

In a recent publication Gent and Vickroy (1966) state that the decrease of A with increasing cross-linlcing does not occur in samples cross-linked in the molten state. Careful inspection shows, however, that this statement is not sustained by the results which they present. In fact, their results on samples cross-linleed in the amorphous state do show the effect to a

similar extent to the result reported here and previously. Their tentative suggestion as to the cause of the effect therefore loses its point.

From the earlier work of Saunders (1956, 1957) the quantity A»* obtained by extrapolating A to 1 = 0 will be taken to be characteristic of the Gaussian network. c

Because of the scatter in Figures 1 and 2 the extrapolation is not easy and has been made by eye treating, as far as possible, the vjhole group of points as a family. In the case of Hostalen G.S., Figure 3, the scatter was

similar in magnitude but in vie\7 of the larger number of points the best

straight lines were fitted through the points according to the method of least squares.

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three poljiiiers in Table k. In the previous publication a value of

A» = 7'5 X 10"^*cm^ was suggested for irradiated samples of a low density polyethylene (Alkathene) at 130°C and a linear polymer, polymcthylene, at l80°C. These values compare well with tha values presented here for the irradiated DYNIC, low density, and HIFAX I6OO, high density, samples (see figure 6 ) .

Dependence_of stress-optical grogerties on temperature

The temperature dependence of A for the DYNK and HIFAX samples is shoim in Figures k and 5. With the exception of the 20 m.rad. sample of HIFAX, the results show a systematic decrease of A with increasing temperature. The behaviour of the 20 m.rad. Hifax sample cannot be explained. It has been

talcen to be a gross experimental error and the offending resxilts have been ignored in discv.ssion. Unfortunately a repeat of the measurements on that sample was not possible. It is felt that the two offending points do not invalidate the general results.

Figure 6 shows the variation of A^ with temperature for sill tliree polymers.

Also incorporated are the earlier results, Saunders (1956, 1957) for a low density polyethylene and polymcthylene and some results on a linear polyethylene quoted by Gent and Vickroy (1966). Tliese last results are obtained on samples cross-linked in the amorphous state and it is interesting to note that the dependence on temperature agrees well with that shown by our Hostalen samples; it should be noted, hoirever, that these values of A have not been extrapolated to zero cross-linking

so that the agreement in magnitude of A must be regarded as of limited quantitative significance. Inspection of Gent and Vickroy's other results suggests that extrapolation would increase the values at all temperatures by about 10^.

Discussion

According to equation (9) the experimentally observable quantity A is given by

A = K 3 K ! 0 ^ - P ^ )

in which K3 and Kx are functions of the parameters describing link-to-link correlation in direction in the ' free' polymer chain and (PP-Pm) is the difference in principal optical polarisabillties for a sin^e link in that chain.

Previous workers have compared the values of A obtained experimentally with values of (PJJ-P ) calculated for monomeric units of known structvire

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. 9

-using tables of bond polarisabillties given in the literature. The calculation involves the assumption that bond polarisabillties may be added tensorially to yield a value of (P-t-Ppi)* ^^^ optical anisotropy of the monomeric unit. Since for a chain óf randomly joined links

K3K1 = 1 . A has been regarded as the optical anistropy of 'the equivalent random link' and the quantity

as the number of monomers per equivalent random link, a measui« of chain ' stiffness' or ' flexibility' .

The best value to be assigned to (^-Pm) has been a matter of some discussion. Denbigh (19^0) published a table of values of bond polaris-abillties, based on measurements made in the gaseous phase, which are the usual basis for calculation of (P^-3jjj). However, there are grounds for criticism of Denbigh' s values and a good general discussion has been given by Volkenstein (1963, p.39l)' In particxiLar, doubt has been cast on the large anisotropy attributed by Denbigh to the C - C single bond. Bunn and Daubeny (195^)^ as a result of measurements on paraffin single crystals, s\3ggested a much reduced value for this quantity and discussion has ensued on which value was the most appropriate for use in work on polymers in the elastomeric state. Saunders (1957) argued in favour of the lower value whilst more recently Volkenstein (1965) and other workers have favoured the higher values. The two different valties lead to significantly different values for (P^-P^) for the - CH2 - unit. Denbigh's values give (P^-8 )_= ik.k X 10"^^cm"^ whilst Bunn and Daubeny's value gives (g^-B ) = 5'1 x 10"^^cm^. In both cases the same values of bond polarisabillties of the C-H bond are adopted. It wovild, therefore, be desirable to have an independent method of estimating (pl£,-Pjjj) from meas\u"ements on polymers in the rubber-like state. Such ah estimate can, in the case of polyethylenes, be obtained in principle from the results presented here.

In order to achieve this it is necessary to have information on the form of K3K1 in eqtiation (9). Adopting the rotational isomeric model for the polyethylene chain and taking the angles to be tetrahedral (see Volkenstein

(1965) for a detailed discussion) one can talce Ki = •/2(2€+l)/3 in which e = expU/RT, U being the energy difference between gauche and trans conform-ations of successive links of the chain. This expression has been obtained by many workers and is well authenticated. The form of K3 is, however, in

some doubt. Kubo (19^9) first obtained the relation

V 1 10€^+l4g-9 ^ 3 = 8 (2e+l)^

w h i l s t a recalcuJLation by Sack ( 1 9 5 6 ) , a c c o r d i n g t o Kubo's method, y i e l d e d

3 8 € ^ + 8 € - l ^ 3 = 8 ( 2 e + l ) ^ •

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Volkenstein (1963) p . U13 quotes r e s u l t s of c a l c u l a t i o n s by Gotlib which

may be reduced to the form K3 = g "Vpe+'i)^ ^ ° ^ °^^ p a r t i c u l a r model.

There is thus some doubt as to the best form for the quantity K3K1

in equation ( 9 ) . For convenience we w i l l i/rite D = K1IC3 and fiurther

r> 1 0 £ ^ l 4 e - i 3e^+8£-l , ^ 12g^+12e+l

^K = 4(2e+l) ' \ = J|(2e+l) ^^"^ ^V = " i^(2€+l)

which are the expressions for D according to the r e s u l t s of Kubo, Sack and

Volkenstein r e s p e c t i v e l y .

We may now w r i t e :

logiolO^^A = logioD + logiol0^*O^-Pj^)

and since (P^i-Pm) i s e s s e n t i a l l y independent of temperature, the temperature

dependence of A occurs in the terms involving D which may take one of the

forms quoted above.

In flgure 7 the experimental \'alues of logiolO^^A are p l o t t e d versus

-and compared with the calculated values of D for the three forms quoted

for each of three chosen values of U, the gauche - trans energy difference.

The values of U diosen are 2,500; 1,150 and 5^0 cals/mole r e s p e c t i v e l y .

The l a r g e value corresponds approximately with that s-uggested by Saunders

(1957) from s t r e s s - o p t i c a l woi-k a t a single temperature. The small value

corresponds to that suggested by Flory, Hoeve and C i f f e r i (1959) from s t r e s s

-temperature measurements. The intermediate value i s t h a t best f i t t i n g the

r e s u l t s here quoted. I t can be seen in Figure 7 that the calculated r e l a t i o n s

between logio^ and — are sensibly l i n e a r over the range covered, in a l l c a s e s .

The three d i f f e r e n t forms of D give, for each value of U, a set of e s s e n t i a l l y

p a r a l l e l l i n e s but the slope of that s e t v a r i e s with U. I t w i l l be seen also

t h a t the experimental r e s u l t s may be taken to be on l i n e s sensibly p a r a l l e l

to t h e l i n e s for D with U = II50 cals/mole. They cannot be nearly so well

represented by l i n e s p a r a l l e l to those for D with U = 2,300 or 5^0 cals/mole.

I t i s suggested therefore t h a t the approximate value for the trans - gauche

energy difference i s ca 1,150 cal.s/mole.

The v e r t i c a l separations between the l i n e s througli the e:cperomental points

and the p a r a l l e l calculated l i n e s for logigD i . e . those for U = 1,150 cals/mole,

gives therefore the value loGiQlO^'*(P!^-Pjn) in each case, and hence the o p t i c a l

anistropy for a single monomeric u n i t . Since in Figure 7 the experimental

points do not a l l l i e on the same l i n e there w i l l be a different value of

(^•^"^m) -^°^ ea.ch. polymer. There w i l l also be d i f f e r e n t values according to

which of the three e:rpressions for D is used. The f u l l set of nine values

of (P{/-Pm2 i s given in Table 5. These values range from 10 x lO'^^cm"' to

19.5 X 10 ^^cm^ and are to be compared with the values llf-.i)- x 10"^^cm3

according to Denbigh' s values of bond p o l a r i s a b i l l t i e s and 3 -1 ^ 10 ^^cm-'

according to the values of Bunn and Daubeny.

(14)

- 11

It is unfortunate that the theoretical form of D is not better established but it may be taken, at least from the results on the two linear polyethylenes, that Denbigh' s values, involving the high C-C anisotropy, are to be preferred to those of Bunn and Daubeny for poly-ethylene in the elastomeric state.

Conclusions

We may conclude therefore that accepting any of the quoted forms for D gives a value for U considerably greater than that suggested by Flory and co-workers from stress-temperature measiirements. The method of obtaining U described here has the merit that one does not need so rigorously to approach the equilibriimi state during measurement as in the stress-temperature measurements since it is well established, and was confirmed in supplementary measvurements during this work, that the value

A n

for — remains constant dxjring the approach to equilibrlxm and also during the early stages of degradativc breakdown of the sample. This latter effect is important in measurements which have to be made at temperatures above the melting point.

Neither this work nor the stress-temperature work takes account of 'Mooney-type' deviations from the behaviour predicted by simple theoiy. Such deviations undoubtedly exist in cross-linlced polythenes as reported by Gent and Vickroy (1966) and as found by us in subsidiary experiments not reported here. It is found, hot/ever, that ~ is sensibly independent of extension ratio and one must conclude therefore that ootli the An vs X and t vs X relations are similarly affected. It is tempting therefore to infer that the results based on stress-optical measurements will be less affected by these departures than those based on direct stress-strain measurements.

The uncertainty in the form of D detracts from the analysis of the stress-optical measurements but it shoiild be noted that all the forms of D give essentially similar values for U. In contrast they give a range of values for {P^-^xa) >'liich in general favour the bond polarisability values of Denbigh rather than Bunn and Daubeny. It is Interesting to note that whatever the form of D the values of (P^-P,^) for the linear chains (Hostalen and Hifa::) are greater than that for the branched chain ( D Y N K ) although there are significant differences between tlie linear polymers.

References

1. Bianchi, Leutzel and J. Polym. Sci., g j , 561. Price (1958)

2. Bunn and Daubeny (195^^) Trans. Farad S o c , ^ , 1175»

5. Ciferri, Hoeve and J. Am. Chem. S o c , 8j, 1015. Flory (1961)

(15)

k.

5.

6.

7.

8.

9.

10.

11.

12.

Denbigh (l9^0)

Flory, Hoeve and Ciferri (1959)

Gent and Vleieroy (1966)

Kubo (19^9)

Mullins and Thomas (1965

Savinders (1956)

Saunders (1957)

Treloar (19^7)

Treloar (1958).

13. Volkenstein and Ptitsyn (1955)

14. Volkenstein (I965)

Trans. Farad. Soc., ^ 6 , 936.

J. Poljm. Sci., 2h:f 557>

In press. J.Appl.Polym.Sci.

J. Phys. Soc. Japan, h, 319.

'The Chemistry and Physics of Rubber-Like Substances' ed. Bateman, p. 175^ publd. MacLaren, London.

Trans. Farad. S o c , ^ , li)-25.

Trans. Farad. S o c , ^ , 86O.

Trans. Farad. S o c , k^,

277-'Physics of Rubber Elasticity' publd. Clarendon Press, Oxford.

Zhiir. Teldi. Fiz. 2^, 6k9.

' Conf igijrational S t a t i s t i c s of Polymeric

Chains' publd. I n t e r s c i e n c e , New York.

(16)

15 -TABLE_1 S t r e s s - o p t i c a l p r o p e r t i e s of DYNK low d e n s i t y p o l y e t h y l e n e c r o s s - l i n k e d by e l e c t r o n i r r a d i a t i o n Nominal sample dose m . r a d . 130°C 20 40 80 l 6 o 145°C 20 40 80 l 6 o l 6 o ° c 20 40 So l 6 o 190°C 20 40 80 l 6 o 1 220°C 20 40 80 160 10^ f cm^/kg 2 . 0 1 1.89 1.90 1.69 1.72 1.77 1.65 1.47 • 1.56 1.61 1.59 1.55 1.42 1.55 1.29 1.11 1.30 1-27 ,, ( 1 . 0 4 ) ' ^ ( .95) .97

ft"

Lxa.iJ

kg/cm" x = i > 1.45 5.75 7.00 1 4 . 1 0 1.70 5.86 7.45 1 5 . 5 0 2 . 6 4 3 . 8 0 7.20 1 5 . 8 0 2 . 5 8 5 . 5 0 7.55 1 6 . 4 0 5 . 0 0 5.95 8 . 1 4 1 7 . 1 0 p gm/cm^ 0.856 0.849 0.846 0.845 0 . 8 5 5 " 0.840 0.842 0.845 0 . 8 2 0 0.825 0.829 0.858 0.786 0 . 7 9 7 0.815 0.352 0.755 0.776 0.795 0.324 fi (H2+2)2 0.0846 0.0849 0.0850 0.0851 O.0847S 0.0855 0.0852 0.0851 0.0861 0.0859 0.0S57 0.0855^ 0.0874 0.0870 0.0865 0.0856 0.0887 0.0878 0.0871 0.0859 lOf^A cm-^ 6 . 9 0 6 . 5 1 6.55 5.85 6 . 1 4 6.55 5.92 5.25^ 5 . 8 6 6 . 0 3 5.94 5.02 5.79 5.47 5.19 4 . 4 5 5.72 5.^9 ,, ( 4 . 5 0 ) " ( 4 . 1 1 ) 4.15 J 10* M c gm"-"-0 . 5 gm"-"-0 1.29 2 . 4 2 4 . 8 9 0 . 5 6 1.29 2 . 6 6 5.18 0 . 8 8 1.25 2 . 5 7 5.15 0 . 8 4 1.12 2 . 5 0 5.02 0.95 1.22 2.45 4 . 9 7 * Difficult to assess

(17)

14

-TABLE 2(a)

ËïïSf Sr2PÏiS?i_Pr2P£ï*i£S_2?_?ïl!^_l^'-'0 high density

polyethylene cross-linked by electron irradiation

1 Nominal I sample j dose m . r a d . 150^C 20 40 80 l 6 o 145°C 20 40 80 l 6 o l6o°C 20 40 80 i 6 o 190°c 20-^=-4 0 80 i 6 o ' 220°C 20-=^ 40 80 l 6 o 1 0 * ^ t cm-/kg 2.16 1 2 . 1 4 i 2 . 1 4

1 1.36

2 . 1 6 2 . 0 6 1.96 1.61 1.88 1.84 1.68 1.55 2 . 1 6 1.62 1.46 1.12 2 . 0 0 1

1.51 1

1.15 i 0.45 ! t uX2 i J -X -j^J leg/cm*^ 1 ^ X=li gm/cm"' i i 1.75 4.20 7.15 1 4 . 5 0 0.85 5.95 6 . 6 0 1 5 . 0 0 1.55 4.55 8.25 1 6 . 0 0 1.51 4.25 7 . 4 0 1 9 . 0 0 1.10 5 . 6 0 1 0 . 5 0 2 0 . 8 0 0 . 8 6 0 ' 0 . 8 4 8 0.846 0.842 0.852 0.855 0.840 0.844 0.818 0.810 0.855 0.823 0 . 7 7 1 0.773 0.805 0.809 0.725 0.742 0.779 0.796 ! n ! (n2-h2) = 1 1

1 0.0345

1 0.0849 0.0850 0.0852 ' 0.0856 0.0855 0.0855 0.0851 0.0861 0.0864 0.0850 0.0854 0.0880 0.0877 0.0867 0.0865 0.0899 0.0891 0.0877

0.0870 1

j lO^^A j cm"' j 7.40^ 1 7.57^ 7 . 3 8 -{ 6 . 4 5 ^ 7.732 7 . 4 l 2 7.05^ 5.76^ 7.05® 6.952 6.21"^ 5.02"^ 3 . 8 6 2 6 . 6 2 * 5.902 4.512 8 . 9 2 7 5 . 7 9 * 5.00^ 4 . 1 0 * j 1 10* i i M j 1 c gm'^ 0 . 5 9 ^ 1-^5° 2 . 4 7 ° 5-05* 0.28® 1.558 2.22

4.55° 1

.51^ 1.46S 2 . 7 0 5.28 .45* 1.59^ 2 . 5 4 2 6.00 .46* 1.16 5 . 2 4 6.24

(18)

15

-TABLE_2(_b)

Stress-optical properties of HIFAX 1600 high density

polyethylene cross-linked by electron irradiation

Nominal sample dose 80 m.rad.

Temp.

"C

i4o

150

160

170

1 180

190

200

210

220

250

240

250

T

413

425

455

445

455

465

475

483

495

505

515

525

1 0 * ^

cm2/kg

2.16

2.00

1.39

1.75

1.61

1.54

1.55

1.45

1.54

1.52

1.20

1.12

P

gm/cm^

.864

.858

.850

.844

.857

.852

.325

.819

.312

.806

.800

.794

n

(£2+2)^

0.0845

0.C«46

0.0849

0.0851

0.0854

0.0856

0.0853

0.0861

0.0364

0.0866

0.0368

0.0871

1 0 ^ f =

8.92

8.46

3.18

7.66

7.29

7.15

7.24

7.00

6.61

6.64

6.16

5.86

102*A

cm^

7.57

7.21

6.99

6.57

6.27

6.14

6.25

6.07

5.75

5.79

5.59

5.14 :

(19)

TA3LE_5

S t r e s s - o p t i c a l p r o p e r t i e s of Hostalen G.S. high density

polyethylene c r o s s - l i n k e d by curing with dicumyl peroxide

1 Nominal p a r t s p e r hundred d i c u p . 156''C ! 10

1 5

1 5

5 5 5 5 182°C 10 10 7.5 7.5 5 5 5

1 I95°c

10 1 10 j 10 7.5 7.5 5

i 5

! 5 3

10* f

cm2/kg 2 . 1 9 2 . 5 7 2 . 2 9 2 . 1 2 2 . 2 2 2.15 2 . 5 7 1.79 1 . 3 1 1.54 1.69 1.95 2 . 0 0 1.82 1.56 1.40 1.46 l . t ó 1.50 1.52 1.60 1.72 1.73 t

.X3.iJ

x = i 9.60 4 . 8 5 4 . 7 9 4.52 2 . 7 5 2 . 5 9 2 . 2 5 1 2 . 6 6 1 1 . 7 2 6 . 2 3 6 . 0 1 1.95 0.95 0.37 1 2 . 5 1 1 1 . 5 4 1 1 . 5 0 8 . 4 1 8.18 4.48 4 . 5 3 2 . 0 8 1.76 p gm/cm-' 0 . 7 9 1 0.792 0.800 0.800 0.768 0.785 0.768 0.796 0.786 0.792 0.792 0.792 0.734 0.773 0.732 0.791 0.782 0.790 0.790 0.779 0.766 0.790 0.790 n (n2+2)^ 0.0872 0.0872 0.0869 0.0869 0.0381 0.0876 0.0881 0.0870 0.0874 0.0872 0.CÖ72 0.0872 0.0875 0.0877 0.0876 0.0872 0.0876 0.0875 0.0875 0.0877 0.0882 0.0875 0.0375 102*A 8 . 1 9 7.90 8 . 5 7 8 . 4 3 8 . 1 9 8.12 3 . 7 4 7.15 7.27 6.16 6.77 7.79 8 . 0 3 7.52 6 . 4 0 5.75 6.05

6.c6

6.14 6 . 2 7 6 . 6 1 7.05 7.07 10* \ gm"^ 3 . 5 4 1.68 1.65 1.56 0.93 0 . 3 4 0.30 4.12 5 . 8 7 2 . 0 6 1.97 0.64 0 . 5 1 0.29 5 . 9 9 5 . 6 9 5 . 7 5 2 . 7 0 2 . 6 2 1.46 1.45 0.67 0.53 — - - - — - '

(20)

17

-Ti\BtE 5 (Contd.)

Nominal

p a r t per

hundred

dicup.

220^0

10

10

10

5

7.5

7.5

7.5

7-5

5

25822

10

10

3

5

10

5

5

7-5

7.5

7-5

5

5

10*

f

Xf

cra2/kg

1.44

1.49

1.51

1.42

1.50

1.53

1.52

1.51

1.56

1-51

1.27

1.59

1.31

1.35

1.51

1.44

1.36

1.40

1.44

1.23

1.57

t

x=i

10.66

9.52

8.84

5.35

2.84

2.34

2.77

2.67

1.50

11.56

11.49

10.89

10.71

10.55

9.84

6.27

5.68

5.40

2.61

1.53

1.52

p

gm/cm-'

0.772

0.794

0.749

0.729

0.742

0.744

0.742

0.744

0.754

O.733I

0.7551

0.7337

0.7337

0.7518

0.7262

0.7193

0.7350

0.7530

0.7371

0.7282

O.72S2

n

(n2-H2)2

0.0880

0.0339

0.0339

0.0896

0.0891

0.0891

0.0891

0.0891

0.0894

0.0S95

0.0895

0.0894

0.0894

0.0887

0.0898

0.0900

0.0895

0.0895

0.0893

0.0897

0.0897

1024A

6.30

6.57

6.65

6.52

6.62

6.77

6.72

6.70

6.94

6.09

5.95

6.45

6.07

6.22

6.12

6.76

6.54

6.52

6.69

6.29

7.30

10*

M

c

gm'^

3.50

2.93

2.82

1.76

0.92

0.91

0.89

0.36

0.49

3.61

5.58

5.39

3.34

3.21

3.10

1.99

1.15

1.06

0.81

0.43

0.41

(21)

TABIE_4

The temperature dependence of A for a l l three polyraers

IT

!°K

405

4l8

|455

465

493

10^ T 2.48 2.59 2.51 2.16 2.05 HIFAX l600 102*A^ 00 cm-'

7.9

7.9

7.i^5

7.0

6.2

logiol02*A^

.8976

.8976

.8722 .8451 .7924 1 0 2 ^ cm^

7.1

6.6

6.5

5.9

5.65

DYNK 1

10giol02*i^ 0.3515 0.8195

0.7993

0.7709

0.7521 T °K

429

455

466

495

511

10^ T 2.53X 2.20 2.16 2.05 1.96 HOSTALEN GS 102*A^

cm^

8.4

8.2

7.2^

6.95

6.9

Ic^iolO^X

.9245

.9138

.8603

.8420

.8389

TABLE_5

The o p t i c a l anisotropy (P^-P^,) for a single

~_GS2_r_ïï2S2ïïê2riQ_üiJi£_£25iEiiJï'S§_l!£Qïï_fisüï;i_7,

Polymer DYNK HIFAX 1600 HOSTALEN GS •• — - — j 1025(M,-PKI) cm-^

from Ds

15.1

17.4

19.5

from Dk

11.9

15.3

15.1

from Dv 1

10.0

11.5

12.9

(22)

1 ,24

Cm

10""* A

3 • U 5 °C €) 160 °C O 190 °C X 2 2 0 °C 1 10 M l

FIG. THE DEPENDENCE OF A ON DEGREE OF

CROSS-LINKING FOR DYNK AT VARIOUS TEMPERATURES

10

6 7

Mc

FIG. 2 THE DEPENDENCE OP A ON DEGREE OF CROSS-LINKING FOR HIFAX 1600 AT VARIOUS TEMPERATURES

(23)

7 8 7 6 8 7 6 7 6 5 7 6 5 2 2 0 ' C ï'ia. 3 10*

THE DEPENDENCE OF A ON DEGREE OF

(24)

10^* A

o

9 X ^^"^«.«^ ' ~ =

-I

Aoo 20MR 40MR 80MR 160 MR ^ 100 150 200 2 50

Temperature "C

FIG. 4 THE TEMPERATURE DEPENDENCE OF A AND A^ FOR DYNK

10^* A

8

o

D A X • € Aoa 20 MR 40 MR 80 MR PART B 80 MR 160 MR 100 150 200 300 a,

Temperature C

FIG 5 THE TEMPERATURE DEPENDENCE OF A AND A „ FOR

(25)

flb

8 0

-7.0

6 - 0

D A for a linear p.c. GENT and VICKROY

150 200 250

Temperature °C

FIG. 6 THE TEMPERATURE DEPENDENCE OF A « FOR THE DYNK. HIFAX 1600 AND HOSTALEN G . S . ALSO INCLUDED ARE THE RESULTS OF GENT AND VICKROY (1966) ON A LINEAR POLYETHYLENE CROSS-LINKED IN THE AMORPHOUS STATE AND THE RESULTS FOR ALKATHENE, POINT 1, AND POLYMETHYLENE POINT 2 FROM SAUNDERS (1956)

(26)

1' 4 log^Q D or 1 - 3 log ( A I O ^ ' ^ ) ^10 ^ ' 1- 2 1 • 1 1- O O • 9 O- 8 O -7 O • 6 O • 5 O • 4 O • 3 O- 2 O • 1 U=2300 U=1150 U= 5 4 0 O A ^ for HIFAX D A . , for HOSTALEN A Aoo 'or DYNK

X A for a linear p.e. GENT and VICKROY

I I I I

1'9 2 0 2-1 2-2 2-3 2 4 2'5

10^

FIG. 7 EXPERIMENTAL RESULTS COMPARED WITH THE CALCULATED VALUES FOR LOG^^D ACCORDING 1 THE EXPRESSIONS OF SACK, VOLKENSTEIN, KUBO,

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