A flow birefringence study of stresses in sheared polymer melts

A FLOW BIREFRINGENCE STUDY OF STRESSES

IN SHEARED POLYMER MELTS

A FLOW BIREFRINGENCE STUDY OF STRESSES

IN SHEARED POLYMER MELTS

^ o

fililllii

« * ' f f i f t \ B | | l l

A FLOW BIREFRINGENCE STUDY OF STRESSES

IN SHEARED POLYMER MELTS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN

DE TECHNISCHE WETENSCHAPPEN AAN DE

TECH-NISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE

RECTOR MAGNIFICUS PROF. IR. L. HUISMAN,

VOOR EEN COMMISSIE AANGEWEZEN DOOR HET

COLLEGE VAN DEKANEN, TE VERDEDIGEN OP

WOENSDAG 8 DECEMBER 1976 TE 16.00 UUR

DOOR

FRANCISCUS HERMANNUS GORTEMAKER

WERKTUIGBOUWKUNDIG INGENIEUR

GEBOREN TE DENEKAMP

KRIPS REPRO

MEPPEL

, 0 U G 8 8 i

'

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR

CONTENTS

1 GENERAL OUTLINE AND MOTIVATION 7

1 1 Preamble 7

1.2 Simple shear flow 8

1.3 Definitions of the tensors used 9

1.3.1 Stress tensor 9

1 3 2 Kinematic tensors 11

1.4 Rheological equations of state 13

1.5 The stress-optical rule 18

1 6 Some further considerations 19

1.7 References 19

2 DESCRIPTION OF APPLIED TECHNIQUES 21

2.1 Cone-and-plate viscometer 21

2.2 Capillary viscometer 22

2.3 Dynamic viscometer 24

2.4 Measurement of the birefringent effect 26

2.4.1 Ehringhaus compensator 27

2.4.2 The compensator according to De Senarmont 28

2.5 The modulation of the birefringence a tool for the accurate

measurement of extinction positions and phase differences 29

2.5.1 The modulator 29

2.5.2 Measurement of the extinction angle 30

2.5.3 Measurement of the birefringence according to

De Senarmont 31

2.6 References 33

3 A RE-DESIGNED CONE-AND-PLATE APPARATUS FOR THE MEASUREMENT OF THE

FLOW BIREFRINGENCE OF POLYMER MELTS 35

3 1 Introduction 35

3.2 Description of the apparatus 36

3.3 Materials and rheological characterization 41

3.4 Kinematics 44

3.5 Results and discussion 49

3.6 Appendix 58

4 FLOW BIREFRINGENCE OF POLYMER MELTS. CALCULATION OF VELOCITY AND

TEMPERATURE PROFILES IN A CONE-AND-PLATE APPARATUS 61

4.1 Introduction 61

4 2 Theory 62

4.3 Results and discussion 68

4.4 Appendix 74

4.5 References 75

5 FLOW BIREFRINGENCE OF POLYMER MELTS APPLICATION TO THE

INVESTIGATION OF TIME DEPENDENT RHEOLOGICAL PROPERTIES 77

5.1 Introduction 77

5.2 Theory 78

5.3 Sample and rheological characterization 83

5.4 Comparison of Lodge's rubberlike liquid theory with flow

birefringence measurements 88

5.5 Appendices 93

5.5.1 Appendix I 93

5.5.2 Appendix II 94

5.6 References 96

6 BIREFRINGENCE BUILD-UP AND RELAXATION IN A SHEARED POLYMER MELT:

RANGE OF VALIDITY OF LODGE'S RUBBERLIKE LIQUID THEORY 97

6.1 Introduction 97

6.2 Coaxial cylinder apparatus 99

6.3 Theory 102

6.4 Rheological characterization of the sample 106

6.5 Comparison of predictions according to Lodge's rubberlike

liquid theory with flow birefringence data 112

6.5.1 Stress growth after the sudden imposition of a

constant rate of shear 112

6.5.2 Stress relaxation after cessation of steady shear

flow 116

6.6 General discussion 121

6.7 References 123

7 TABLES OF EXPERIMENTAL RESULTS 124

7.2 Time dependent flow birefringence measurements 127

SUMMARY 132

SAMENVATTING 135

LIST OF SYMBOLS 138

CHAPTER 1

GENERAL OUTLINE AND MOTIVATION

1 1 PREAMBLE

Whereas classical fluid dynamics is traditionally restricted to the

study of the behaviour of simply behaving (Newtonian) liquids or gasses in

more or less complicated flow fields, rheology (which, as a matter of fact,

IS a synonym to fluid dynamics) is concerned with the more complicated flow

behaviour of materials as such molten polymers, polymer solutions, emulsions,

suspensions etc. The flow behaviour is characterized by the dependence of

stresses on deformations occurring in the material or visa versa

flathe-matically, this dependence is formulated m a rheological equation of state

or constitutive equation Very well known examples of such equations are

Hooke's law for a purely elastic material and Newton's law for a purely

viscous material.

A material is called an elastic solid (LODGE (1)) if i) it has an

equilibrium shape at zero stress, ii) if the material is kept in a shape

different from the equilibrium shape, the stress reaches a non-isotropic

equilibrium determined, within an additive hydrostatic pressure, by the

given shape

A material is called a purely viscous liquid if a) the liquid remains

m a constant shape as soon as the stress becomes zero, b) the stress becomes

zero as soon as the liquid is held in any constant shape A viscous liquid

obeys Newton's law, which calls for proportionality between stress and rate

of deformation.

The two definitions just given for an elastic solid and a viscous

li-quid can only be applied to limiting cases Real materials, m general, do

not fall readily into one or the other of the above-mentioned categories At

least one additional parameter is needed to describe material behaviour more

realistically This extra parameter is time. As a general rule, the faster

the deformation the closer the response is to that of an elastic solid,the

vis-cous liquid Fast and slow deformations are notions relative to some natural

time, T, of the material This natural time is thought to be related to the

rates of spontaneous diffusion of the material's molecular and atomic

con-stituents For usual fluids like water, the natural time is very short, of

the order of 10 seconds (2) and hence, for the most purposes, these

fluids are considered as being viscous For a material like glass, the

na-tural time IS very long, and consequently, such a material is bracketed with

elastic solid Materials with natural times of the order of the duration of

daily events (seconds, minutes, hours) are qualified as visco-elastic

mate-rials The polymer melts of this investigation fall into this class

Polymer melts and polymer solutions show also some unusual behaviour

during flow One example for such a behaviour is the generation of the

"WEISSENBERG effect" (3) If a rotating cylinder is dipped into a polymer

solution or melt, the liquid climbs the wall of this cylinder Another

example is given by the so-called "die swell" (BARUS effect) (4) This

ef-fect occurs when a suitable liquid emanates from a capillary The diameter

of the extrudate can be 1 to 5 times that of the capillary A positive

extrudate swell can only be explained by the presence of considerable

nor-mal stresses in the fluid, as induced by the flow Another characteristic

effect occurring with polymer melts and some polymer solutions is "melt

fracture" (5) The extrudate from the capillary becomes distorted if the

shear stress at the capillary wall exceeds a critical value

1 2 SIMPLE SHEAR FLOW

Simple shear flow is an important class of flow and forms the subject

of this thesis In a right handed Cartesian coordinate system, the

coordi-nate axes represent the following characteristic directions 1 - direction

of flow, 2 - direction of the velocity gradient, 3 - neutral direction

The coordinates of a fluid particle at some previous time t' and at

the present time t (of observation) are given by (x', x', x') and

(x , X , X ) , respectively If, as an example, a time independent

("con-stant") rate of shear q is assumed, the position of the particle at the

time t of observation depends in the following way on that at the previous

time t'

x„ = x'

2 2 (1.2)

""3 = ""3 (1.3)

The shear accumulating between t' and t is given by y(t,t') = (t-t')q.

1.3 DEFINITIONS OF THE TENSORS USED

1.3.1

Stress Tensor

The stress tensor is defined as follows.

"11 '^21 31 12 22 32 13 ^23 33 (1 4)

For an elastic solid (e.g. a rubber) it can easily be shown (1.6) that the

stress tensor is symmetrical, i.e.:

E = E

Unfortunately, this cannot be proven straightforwardly for any fluid

mate-rial. Also experimental evidence can only be indirect for a fluid (6).

How-ever, for a rubberlike fluid like a polymer melt we probably may assume

sym-metry of the stress tensor, at least for the regime of not too high rates of

shear. As a consequence, the number of independent stress components is

re-duced from nine to six.

From the symmetry of the field of simple shear deformation for an

isotropic material it follows that p = p = 0 and p = p = 0.

Conse-quently the most general state of stress for an isotropic material in simple

shear is represented by the following Cartesian stress components.

•^ij 11 12 P l 2 P 2 2 0 0 0 p 33 (1.5)

The quantities p , , , p_„ and p are normal stresses and P.„ = P^^ Is the

shear stress. In an incompressible m a t e r i a l , the state of stress is

determined by the deformation (history) except for an additive hydrostatic p r e s

-sure, the latter being dependent on certain boundary conditions. Therefore

values of seperate normal stress components do not have a direct rheological

meaning. From a rheological point of view only the values of the differences

between the normal stress components and the shear stress (the "deviatoric"

components of the stress tensor) are of interest. In other w o r d s , there are

only three independent stress quantities of rheological significance, namely

11 22' •^22 33' '21 (1.6)

p follows from the difference of the first and the second one. where p, - p„„ is called the first normal stress difference and p - p

11 22 *^22 33 the second normal stress difference. The third normal stress difference

Pll •

The stress tensor can also be represented by means of a

stress-ellips-oid. In a Cartesian coordinate system formed by the three principal axis of

this ellipsoid, this tensor has only diagonal components. All components

with mixed indices are equal to zero. It can easily be seen that, for the

above specified shear, one of the principal axes of this ellipsoid (the

Ill-axis) coincides with the 3-direction As a consequence, the other

prin-cipal axes (the I- and II axes) must lie in the plane of flow, i.e. the

1,2-plane. If the principal axis making an angle \ smaller than fortyfive

tv,=qy

Fig. 1.1 Laboratory coordinate system:

X ... direction of flow (also

1-direc-tion), y ..• direction of velocity

gra-dient (also 2-direction), I,II ...

prin-cipal directions of stress,

x •••

orientation angle of stress ellipsoid,

V ... velocity (in x-direction), q ...

velocity gradient (shear rate).

degrees with the direction of the stream lines (see Fig. 1 1) is defined as

the first principal axis, then the following equations are obtained by a

princi-pal axes to the axes of the laboratory system:

Ap sin 2X^ = 2 p^^ (1.7)

Ap cos 2x^ = P,, - P22 (1.8)

''ill ^33 (1.9)

where Ap = p - p is the difference of the two principal stresses in the

plane of flow. If eq. (1.8) is divided by eq. (1.7), an expression for the

orientation angle X °f the stress-ellipsoid is obtained. It reads:

cot 2x Pll " P22

2 p (1.10)

21

1.3.2 K%nematvc Tensors

The Cauchy Deformation Tensor

In a Cartesian coordinate system the Cauchy tensor is defined in the

following way: 3 3x' dx' C = T — 5 . - ^ ij , 3x. 3x. s=l 1 J (1.11)

where x' is the position vector of a material point before deformation and

X the corresponding vector after deformation. The components of the Cauchy

tensor are for a simple shear deformation y.

ij 1 Y 0 -Y l^Y^ 0 0 0 1 (1.12)

As is well-known, C describes the deformation of a body in terms of the

change of the square of the distance between two neighbouring material

The Finger Deformation Tensor

In a Cartesian coordinate system the Finger deformation tensor is

de-fined in the following way

3 3x 3x B = Z - i

^^ s=l 3x' 3x'

(1.13)

The Finger deformation tensor is equal to the inverse of the Cauchy

defor-mation tensor:

B _ = C ij iJ

-1

(1.14)

The components of the Finger deformation tensor are for a simple shear

de-formation Y-Id l+Y Y 0 Y 1 0 0 0 1 (1.15)

The Finger tensor describes the deformation m terras of the change of the

square of the distance between two neighbouring parallel material planes.

The Rate of Strain Tensor

In a Cartesian coordinate system the rate of strain tensor is defined

as follows.

ij 2 \ 3x

1 I ^ \ ' \ _(1.16)

where V^ is the speed of a material point.

The Invariants of the Introduced Tensors

As IS well-known, every symmetric second rank tensor possesses three

ten-sors are considered, these invariants play a certain role in the model

building (construction of constitutive equations),as they can be introduced

as scalar variables into the arguments of suitable functions The second

in-variant of p IS, for example, often used to express the shear rate

dependen-cy of rheological properties.

For an arbitrary symmetric tensor A one obtains:

\ = \ i

* S 2 * S 3 = \ ^

^ Sii

2 2 2 S " ^11^^22 "" ^ 2 * 3 3 "" ^ 3 ^ 1 1 " •*12 " ^23 " ^^31 "

= AjAj,j + A^jAj.jj + AjjjAj

I = det (A ) (1.17)

o 1 J

In particular, the invariants of Q and g are equivalent in the above

stipulated application. The third invariant of the just mentioned tensors

IS unity for incompressible fluids, as it indicates the ratio of the final

to the initial volume.

For incompressible fluids the first invariant of the rate of strain

tensor is equal to zero. For the most important types of flow ( m

particu-lar viscometric flows) of incompressible fluids also the third invariant

of the rate of strain tensor is equal to zero

Therefore in model building always the first and the second invariant

of the strain tensor and the second invariant of the rate of strain tensor

are used.

1.4 RHEOLOGICAL EQUATIONS OF STATE

The rheological equations of state, which have been proposed in the

past, can be subdivided into two groups, viz. the differential and the

in-tegral models. A survey and comparison of the most important models was

given among others by SPRIGGS, HUPPLER and BIRD (7) and by BOGUE and DOUGHTY

(8,9). All known differential models involve special forms of a differential

field of polymers, however, is generally given to the integral models and,

in particular, to the relatively simple single integral models. As a matter

of fact, models derived from the classical theory of rubberelasticity fall

into this category. As the hypothesis, according to which polymer melts

de-rive their mechanical properties from a rubberlike entanglement network, is

generally accepted (11) and possesses a sound physical background,

consi-derations are restricted in this thesis to models of the latter type.

The integrand of a single integral model is formed by a product of a

memory function and a strain increment. The first constitutive equation of

this type, as proposed in the literature, was LODGE'S constitutive equation

for a rubberlike liquid (1). This rheological equation of state has the

same form as the one derived by the same author (12) from the molecular

theory of concentrated polymer solutions. In such a solution a network of

physical entanglements is supposed to be continuously broken down and

re-built under the influence of thermal motion. The concentration of network

junctions which are formed in a time interval between t' and t' + dt' and

which are still existent at a later time t, is represented by a memory

function y(t-t')dt', which decreases as the value of t-t' increases.

According to LODGE's theory, each component of the stress tensor at

time t IS related by the same function iJ(t-t') to the corresponding

compo-nent of the Finger strain tensor at time t'. The following expression

re-sults from this supposition

t

p(t) + P 1 = U(t-t') C"''' (t,t') dt' (1.18)

—CO

where p is the undetermined hydrostatic pressure. This equation represents

a direct generalization of the theory for an ideal rubber (13 - 18). As a

matter of fact, the permanent network of an ideal rubber behaves neo-Hookean

(19), obeying the following constitutive equation.

2 + p 1 = G g~^ (1.19)

The proportionality factor G is the equilibrium shear modulus of the

net-work It IS given by

where V is the number of effective chains per unit volume, k Boltzmann's

constant and T the absolute temperature. In deriving these equations one

has assumed that the deformation of an ideal rubber occurs at constant

vo-lume and with no change of internal energy An empirical extension of eq.

(1.19), as proposed by MOONEY and RIVLIN for non-ideal rubbers, has been

used extensively to correlate experimental results This equation reads

E + P 1 = Cj^ g~^ " *^2 S (1-21)

where C and C are experimentally determined constants.

A modification of LODGE's constitutive equation (1.18) was proposed by

WARD and JENKINS (20). In this constitutive equation a term is included

analogous to the "MOONEY" term (21) in eq. (1.21):

t

E(t) + p 1 = ryj^(t-t') g"^ (t,t') + y^^t-t') C(t,t')l dt' (1.22)

The main effect of the introduction of this term is that the new

constitu-tive equation furnishes not only a first normal stress difference, as eq.

(1.18) did, but also a second normal stress difference (see the

compo-nents of B and C in 1.3 2 ) .

However, as eqs (1.18) and (1 22) are still unable to describe the

experimentally observed dependence of the viscosity of polymeric fluids on

the rate of shear, LODGE (ref (1) p 121 problem 5) suggested that the

second invariant I of the rate of strain tensor may play a role as an

in-dependent variable in the argument of the memory function. Also FREDERICKSON

(22) has made the same suggestion. Based on this assumption LODGE suggested

to SPRIGGS, HUPPLER and BIRD (7) an equation of the following form.

t

E(t) + p 1 = I p(t-t', l2(t')) [(1 + f) S"'''(t,t') + I g(t,t') 1 dt'

—00

(1.23)

where, in contrast to eq (1 2 2 ) , only one memory function occurs. On the

other hand, this memory function is assumed to be dependent on I . Quanti

hypo-thesis' . Equation (1.23) gives the

WARD-JENKINS-FREDERICKSON-LODGE-HOONEY-?)

BIRD (WJFLMB) model .

Another model derived from LODGE's rubberlike liquid model is the

BERNSTEIN-KEARSLEY-ZAPAS (BKZ) model (28), see also (29).

t

E + p i = rp^(t,t') g"-^(t,t') + y^Ct.t') e"-^(t,t').c'^(t,t') j dt'

— 00

(1 24)

In contrast with the assumptions of the WJFLMB model the authors of the

BKZ model propose the use of the invariants of the deformation tensors in

their memory functions.

On these models numberless variations were invented, e g. (30-39) A

description of all these models falls far outside the scope of the present

investigation. In fact, m our opinion the present state of the experimental

techniques is insufficient to scrutinize the predictions in all respects.

To support this statement an example is quoted from the literature.

MEISSNER (40) as well as HANSEN (41), and recently also HACOSKO

and MORSE (42) showed that normal stress response after a sudden start of

shear flow strongly depends on the gap angle of the used cone-and-plate

ap-paratus. The observed delay in response is ascribed by the authors to the

small deflection of the plate necessary for the normal force measurement.

In contrast to WEISSENBERG's original postulation, several authors

(23-27) have indeed found experimentally that the second normal stress

diffe-rence was final and proportional to the first normal stress diffediffe-rence

for a series of polymer melts.

2)

It should be noticed, however, that the investigation described m this

thesis shows that the memory function of eq. (1.23) is not suitable for

polymer melts. In fact, for short lapses of time ( t - f ) it could be shown

that the memory function must be independent of I (see Figures 5.10 and

6.6). This finding is in contrast to the assumption made in eq. (1 23).

Another, already well-known fact, vz. the linear behaviour of polymer

melts within a wide range of oscillation amplitudes (even at high

fre-quencies) fits very well in this picture, i.e. that y cannot depend

In this way, one has never been able to check seriously the usefulness of a

constitutive equation for the description of transient normal stresses. The

flow birefringence technique, however, which is used in the present

investiga-tion, has the advantage that the apparatus can be made sufficiently rigid At

the same time, the sensitivity of the optical measurement can be tuned up

al-most whitout limits. With the aid of this optical technique it is possible to

obtain measurements at extremely low shear rates, where one can expect the

validity of the most simple constitutive equations containing memory

funct-ions which depend only on the Brownian motion and are independent of

invari-ants of the mentioned kinematic tensors. This provides us with the

oppor-tunity to do a step back with respect to what other people wanted to measure

(see also a similar move by LODGE and MEISSNER (43)) So we discard the

in-fluence of the invariants from the constitutive equations.

For the moment, also the second normal stress difference was

disre-garded. So far, there are only a few reliable steady state measurements

(23-27) of this quantity. For several polymer melts WALES (23,25) determined the

second normal stress difference optically as a function of shear rate. A

capillary apparatus was used for these measurements As already mentioned,

a constant ratio between second and first normal stress differences is found.

For various polymer melts this ratio lies between -0 08 and -0.14, i.e.

Pr,o ~ P n , has always the reverse sign of p - p„„. For polymer melts,

mea-surements of build-up and relaxation of the second normal stress difference

seem still inaccessible. For some polymer solutions, however, LEPPARD and

CHRISTIANSEN (27) showed that the normalized first and second normal stress

difference grow and decay at the same rate. This beautiful work was carried

out with the aid of miniature pressure gauges mounted in the rigid walls of

the cone-and-plate apparatus. Unfortunately, these pressure gauges were not

suitable for work at elevated temperatures necessary for polymer melts.

When equations (1.12) and (1 15) are substituted into eq. (1.22) the

following expressions are found for the shear stress and the first normal

stress difference as occurring during simple shear flow:

t

°21 " tJ(t-t') Y(t,t') dt' (1.25)

with

p(t-t') = y^(t-f) - U2(t-t') (1 27)

From these equations it appears that for this type of measurements, it

makes no difference whether one uses the original equation (1.18) according

to LODGE or the more general equation (1.22) (see also OSAKI (44)). The

ex-pressions (1 25) and (1.26) also follow from the theory for a second order

fluid (45,46)

Recently, OSAKI (44) has shown that, for moderately concentrated

so-lutions of extremely high molecular weight polymers, the BKZ model forms a

better basis for the interpretation of transient shear stresses than the

WJFLMB model As polymer melts do not necessarily display the same

proper-ties as the above mentioned solutions, investigations along OSAKI's line

will probably form a part of the future program in our laboratory In this

respect also the work of TSCHOEGL and coop. (48) on the stretching of rubbers

will be taken into consideration

1 5 THE STRESS-OPTICAL RULE

In the present work stress build-up and stress relaxation are

investi-gated with the aid of the flow birefringence technique. For the purpose, use

IS made of the stress-optical relation.

5' = C 2' (1 28)

where n' and g' are the matrices of the deviatoric components of the

re-fractive index and the stress tensors, C being a constant, the so-called

stress-optical coefficient Also this relation is borrowed from the theory

of rubber elasticity, where the following expression is found for C (49):

2TT (n + 2) (1^ - a )

n being the refractive index of the isotropic medium and a - a the

dif-ference in polarizability of the statistical random links, of which the

chain molecules are supposed to be built up.

The same relation is found by LODGE for his rubberlike liquid model

the two tensors, i.e.

X = X„ (1.30)

where x ^s the extinction angle, cannot be considered as a general law for

fluid systems, it has to be checked carefully.

1.6 SOME FURTHER CONSIDERATIONS

The memory function which occurs in eq. (1.25) and (1.26), is also

in-corporated in the respective equations for the dynamic shear moduli, as

given by LODGE'S rubberlike liquid model. A recently published very

conve-nient method for the interrelation of linear viscoelastic functions - see

SCHWARZL and STRUIK (47) - is used in the present work to predict transient

shear stresses and first normal stress differences from the dynamic moduli.

For this purpose the latter quantities were measured as functions of the

circular frequency oj. The mentioned stresses are obtained at t = 1/u. As

already mentioned above, we restrict ourselves to the second order fluid

range. Details of this calculation are given m Chapters 5 and 6.

As Chapters 3-6 of this work have already been published as separate

papers in Rheologica Acta, some of the expressions given in this general

outline will be found again in these Chapters.

1.7 REFERENCES

1) LODGE, A.S., "Elastic Liquids", Academic Press, New York (1964). 2) FRENKEL, J., "Kinetic Theory of Liquids", Clarendon Press, Oxford 1946,

Ch. 4.

3) WEISSENBERG, K., Proceed., 1st Intern. Congr. on Rheology, I 29, North Holland, Amsterdam, 1949.

4) BAGLEY, E.B., S.H. STOREY, D.C. WEST, J. Appl. Polymer Sci., 1^. 1661 (1963).

5) SPENCER, R.S., R.E. DILLON, J. Colloid Sci., £ , 241 (1949).

6) MERK, H.J., Macrorheologie A. Principes van de contmuumsmechanica, Collegedictaat, Delft.

7) SPRIGGS, T.W. , J.D. HUPPLER, R.B. BIRD, Trans. Soc. Rheol., 10, 191 (1966).

8) BOGUE, D C , J.O. DOUGHTY, I.E.G. Fund., 5, 243 (1966). 9) DOUGHTY, J.O., D.C. BOGUE, I.E.C. Fund., 6, 388 (1967). 10) OLDROYD, J.G., Proc. Roy. S o c , A 245, 278 (1958). 11) GRAESSLEY, W.W., Adv. Polym. Sci., 16, 1 (1974).

12) LODGE, A.S., Trans. Faraday S o c , ^ , 120 (1956).

14) KUHN, W , Koll Z , 76, 258 (1936)

15) GUTH, E , H H JAMES, J Chem Phys , 11, 455 (1943) 16) WALL, F T , J Chem Phys , 10, 132 (1942)

17) TRELOAR, L R G , Trans Far Soc , 39^, 36 (1943)

18) GREEN, M S , A TOBOLSKY, J Chim Phys , 1^, 80 (1946)

19) RIVLIN, R S , In Eirich, F R (Ed ) Rheology, Vol 1, 351, Academic Press (1956)

20) WARD, A F H , G H JENKINS, Rheol Acta, 1. 110 (1958) 21) MOONEY, M , J Appl Phys , jA, 582 (1940)

22) FREDERICKSON, A G , Chem Eng Sci , r7, 155 (1962)

23) WALES, J L S , The Application of Flow Birefringence to Rheological Studies of Polymer Melts , Delft (1976)

24) TANNER, R I , 6e Congres de Rheologie, Lyon (1972) 25) WALES, J L S , W PHILIPPOFF, Rheol Acta 12, 25 (1973)

26) CHRISTIANSEN, E B , W R LEPPARD, Trans Soc Rheol , 18, 65 (1974) 27) LEPPARD, W R , E B CHRISTANSEN, AIChE J , 21,, 999 (1975)

28) BERNSTEIN, B , E A KEARSLEY, L J ZAPAS, Trans Soc Rheol , 7, 391 (1963)

29) KAYE, A , Note No 134 of College of Aeronautics, Cranfield, October (1962)

30) LEONOV, A I , G V VINOGRADOV, Doklady Akad , Naok SSSR, 155, 406 (1964) 31) LEONOV, A I , Zh. P II T F (J Appl Mech Techn Phvs ) 4, 78 (1964) 32) LEONOV, A I , A Ya MALKIN, Zh P M T F , 5^, 68 (1965)

33) SIMMONS, J M , Ph D Thesis Sydney (1967)

34) TANNER, R I , J H SHIHONS, Chem Eng Sci , ^ , 1803 (1967) 35) BOOY, H C , Thesis Leiden (1970)

36) CARREAU, P J , Trans Soc Rheol , 16, 99 (1972)

37) PHILLIPS, M , Proceedings of Vllth international Congress on Rheology, p 452 (1976)

38) CHANG, S J , L J ZAPAS, Proceedings of Vllth international Congress on Rheology, p 567 (1976)

39) WAGNER, H H , Rheol Acta, 1^, 133 (1976) and Rheol Acta 15, 136 (1976) 40) MEISSNER, J , J Appl Polym Sci , 1_6, 2877 (1972)

41) HANSEN, M G , Ph D dissertation, University of Wisconsin (1974) 42) MACOSKO, C W , B J MORSE, Proceedings of the Vllth international

Congress on Rheology, p 376 (1976)

43) LODGE, A S , J MEISSNER, Rheol Acta, 12, 41 (1973)

44) OSAKI, K , Proceedings of the Vllth international Congress on Rheology, p 104 (1976)

45) COLEMAN, B D , H MARKOVITZ, J Appl Phys , 35, 1 (1964)

46) COLEMAN, B D , H MARKOVITZ, J Appl Polym Sci , 1^, 2195 (1974) 47) SCHWARZL, F R , L C E STRUIK, Adv Molec Relaxation Processes, 1,, 201

(1967)

48) CHANG, W V , R BLOCH, N W TSCHOEGL, Proceedings of the Vllth inter-national congress on rheology, p 240 (1976)

BLATZ, P J , S C SHARDA, N W TSCHOEGL, Trans Soc Rheol , 18, 145 (1974)

49) TRELOAR, L R G , 'The Physics of Rubber Elasticity (2 nd), Oxford (1958)

CHAPTER 2

DESCRIPTION OF APPLIED TECHNIQUES

2.1 CONE-AND-PLATE VISCOMETER

With such an arrangement the liquid is held between a rather flat cone

and a plate, which rotate relatively to each other around a common axis (see

Fig. 2.1). An essential feature is that the apex of the cone rests exactly

on the plate. In this way a rather narrow gap is formed between cone and

plate. The advantage of this arrangement is that a practically uniform

shear rate is created in the liquid, with which the gap is filled. For a

sufficiently small gap angle ijj one obtains (1) :

(2.1)

where q is the shear rate, w the angular velocity of the cone. In this fi-c

gure the plate is assumed to be stationary. The derivation of this simple

equation is based on the assumption of

closed circular streamlines. In contrast

to an arrangement of concentric

cylin-ders, the curvature of these streamlines

does, in principle, not become

disregar-dable, when the gap is drastically

re-duced. For a discussion of this and

Fig, 2,1 Scheme of the cone-and-plate viscometer:

R .,. radius of cone and plate,

'l'^ • • • S'OP angle,

other problems reference is made to LODGE'S book (1). For the present

pur-pose it suffices to give two equations showing the relations between the

torque M (1,2) and the shear stress p„,, and between the total axial thrust a 21

F and the first normal stress difference p - P„„, respectively:

2 TT R 3

\--3— P2I (2.2)

^

(P,, - Poo) (2-3) a 2 "^11 '^22

where R is the distance of the rim of the cone from the axis of rotation,

subscript 1 stands for the tangential direction with respect to the stream

lines in a point P and subscript 2 for a direction perpendicular to the

stream line and the radius vector in that point. The validity of the second

equation was first proposed by WEISSENBERG (3). It probably holds

suffi-ciently well under the simple condition that the pressure at the open rim

of the liquid is equal to the atmospheric pressure, irrespective of the

pre-cise shape of the rim.

The measurements presented m the following chapters are obtained with

the aid of the Weissenberg rheogoniometer of high perfection, as described

by MEISSNER (4).

In general, measurements with this apparatus are restricted to rather

low rates of shear. At higher rates of shear, no steady state readings are

obtainable, presumably because the open rim of the polymer melt starts to

tear up.

2.2 CAPILLARY VISCOMETER

A ram viscometer (see Fig. 2 2 ) , which was designed by J VAN LEEUWEN

and R. VAN DER VIJGH at the Plastic and Rubber Institute T.N.O (Delft),

was used for the measurement of viscosities in the high shear rate region.

The measured quantities are the volume output per second Q and the pressure

P in the reservoir before the entrance to the capillary.

The volume output per second can be calculated from the ram speed V r and the radius R of the reservoir

r

Q = IT R V (2.4) r r

Fig. 2.2 Saherriatic cross-section of the capillary viscometer: A . . . ram, B . . . bar-rel, C . . . capillary, D . . . reservoir, E . . . hole for pressure transducer, L ... length of the capillary, R . . . radius of the capillary, R . . . radius of the reservoir.

The apparent shear rate at the

capil-lary wall q , is defined as:

1„ = _{a TT R} 4 Q (2.5)

where R is the radius of the capillary. c

When eq. (2.4) is inserted in eq. (2.5)

'the following expression for the apparent

shear rate is found:

4 R 2 V

r r (2.6)

BAGLEY (5) proposed the use of an effective length of the capillary to

cor-rect for entrance effects:

,-1

(2.7)

''w =1 Li! *

"BJ

where n is the BAGLEY correction which slightly varies with shear stress

a

a at the wall but is supposed to be independent of L /R . L is the length w c c c

of the capillary. RABINOWITSCH (6) derived the following expression for the

true shear rate at the capillary wall (no slip condition):

q(R ) c q r d In q ~| ^ 3 ^ a 4 d In a I— w—' (2.8)

For a Newtonian fluid one obtains d In q /d In a = 1 and, as a consequence.

a w

q(R ) = q . Capillaries with an entrance angle of 90 degrees were used. The c a

radius of the reservoir R was 10.2 mm. The capillaries had identical radii r

(i.e. 0.817 mm) but varying lengths (viz. 15.0, 30.0 and 75.0 m m ) . In this

keeping in mind that there exists a unique relation between q and O . a w

Capillary rheometer data, according to eq. (2.8) are in general assumed

to give steady flow properties of the polymer melts. This means that

tran-sient effects are incorporated in the entrance correction.

2.3 DYNAMIC VISCOMETER

The linear viscoelastic behaviour of polymer melts was investigated

with a dynamic viscometer of the concentric cylinder type (see Fig. 2.3). -3

This apparatus covers an angular frequency range from 10 to about 300 rad/

sec. It was designed by K. TE NIJENHUIS, making use of earlier experiences

by several other authors (7-10).

Many other types of apparatuses for the measurement of dynamic

proper-ties of fluid systems have been published, especially for the high frequency

range (up to the mega cycle range) (11,12)

The apparatus, as developed by K. TE NIJENHUIS furnishes accurate data I in the range of medium to very low frequencies, where the

cj.^ Ej phase angle differs little from ninety degrees. Details

I I S of this apparatus will be published by the mentioned

author. Only the general principals will be outlined

here.

The polymer melt is held between concentric

cylin-ders. The inner cylinder is suspended between two

torsi-on wires. The upper torsi-one is ctorsi-onnected to a driving shaft,

whereas the lower, relatively thin one, is mainly

intend-ed as a means to keep the inner cylinder centrintend-ed. The

driving shaft performs a sinusoidal oscillation around

the common axis of the system. If the amplitude of the

iL, /^ oscillation is not too high, the inner cylinder also

7\

/s

[7

'<:

viscometer: h ... height of inner cylinder, D. ... dia-meter of inner surface of the outer cylinder, D-, D„ ... torsion wires, S ... driving shaft, e^, e ... an-gular amplitude of driving shaft and inner cylinder, respectively.

acquires a sinusoidal motion of the same frequency, so that the fluid

be-tween the stationary outer cylinder and the oscillating inner cylinder

un-dergoes a sinusoidal shear The combined influences of the viscoelastic

properties of the fluid, the inertia of the inner cylinder and the torsional

stiffness of the upper wire are responsible for differences in phase and

am-plitude between the driving shaft and the inner cylinder By a photoelectric system the amplitudes £ (of the driving shaft) and £ (of the inner

cy-ao CO

linder), as well as the phase angle <\) between shaft and cylinder are measured If the torsional stiffness of the torsion wire, the geometry of

the cylinders as well as the momentum of inertia of the inner cylinder are

known, the shear moduli of the fluid can be calculated By varying the

angular frequency to of the oscillation one is enabled to investigate the

complex shear modulus as a function of to As is well-known, the real part

of this modulus, which is in ph^se with the shear, is called the storage

modulus G', whereas the imaginary part, which is 90 degrees in advance of

the shear, is called the loss modulus G" In this consideration it is

as-sumed that the deformation is sufficiently small, so that the behaviour of

the fluid is in the linear region It can be shown that this condition is

easily satisfied for polymer melts (10) The shear moduli are obtained from

the following equations D,

- 1 I 2

. - CO (2 9)

G- = _ (2 10)

where E /£ is the ratio of the amplitudes of driving shaft and cylinder, ao CO

D is the torsional stiffness of the wire, I the momentum of inertia of

the cylinder and E a geometric constant, i e

ir h

„ 2 „ 2 D D

1 O

(2.11)

In the latter equation D is the outer diameter of the inner cylinder, D

1 o the inner diameter of the outer cylinder and h the length of the inner

For the interpretation of the measurements only D /E and I/E are

re-quired. These quantities are obtained by measurements with a non-elastic

liquid of known viscosity (n = G"/a-, G' = 0) at various frequencies. Another

method consists in a direct measurement of D and a calculation of E from

eq. (2.11). The value of the momentum of inertia I was calculated from the

resonance frequency of the empty system (9). The inner diameter of the outer

cylinder was 9 00 mm, the diameter of the inner cylinder was 5.00 mm and the

length of the inner cylinder was 50.00 mm. The wires used had diameters of

1.00, 0.75, 0.45, 0.30 and 0.20 ram

As mentioned above, the inner cylinder was centred by a second torsion

wire at the bottom. Apparently, this second torsion wire affects only the

measurement of G'. As this disturbance was considerably less than 1% for all

measurements, it could entirely be neglected.

2.4 MEASUREMENT OF THE BIREFRINGENT EFFECT

In all types of instruments for the measurement of flow birefringence,

to be described in that which follows, a measurement of the created

bire-fringence will be necessary. As this measurement is carried out according

to well-known principles of optics, it will be treated separately in the

section.

There are many ways of measuring optical path differences. A full

des-cription of this subject, however, is beyond the scope of the present work.

In the simplest case the phase difference 5 between two linearly polarized

rays of monochromatic light, vibrating in mutually perpendicular planes,

just amounts to a multiple of 2i, say, 21 m. The emergent light is then ex-tinguished by the analyzer. The pertinent path difference is then given by:

r = -^— . A = m X (2.12)

where ' is the wave length of the light. We chose a wave length of 546 nm.

The birefringence is given by.

An = 7 (2.13)

LI

where L is the path length of the light in the (homogeneously) birefringent

birefringen-ce can be generated by a reasonable shear stress) , m can be found by the

application of a slowly but steadily increasing shear stress and by simply

counting the extinctions ("fringes") , which pass before the final stress

is reached. Under many circumstances, however, one is m need of

measure-ments of path differences which are fractions of one wave length In the

present investigation this was achieved by the use of two conventional types

of compensators a) the EHRINGHAUS compensator (13) which is well known to

the users of the polarizing microscope, and b) the compensator according to

DE SeNARMONT (14), which is useful for very small birefringences

2 4.1 EHRINGHAUS compensator

In this compensator a rotating plate composed of two quartz crystals

IS used This plate is placed into the light beam m series with the double

refracting sample to be measured. The crystals are cut and glued together

in such a way that no phase difference is generated when the compensator

plate is in an exactly perpendicular position to the light beam On

ro-tation of the plate about one of its principal axes, a birefringence effect

is produced By placing the axis of rotation of the compensator

successive-ly m both mutualsuccessive-ly perpendicular extinction positions of the sample, the

subtraction position can easily be found m white light. In this way also

the sign of the birefringence of the sample can be determined With

mono-chromatic light extinction is obtained vhen the sum of the path differences

of sample and compensator is a multiple of 2TT . In normal use the subtraction

position IS used and the method is that of compensation. The relation

be-tween the rotation, read from a scaled drum, and the pertinent phase shift,

is supplied in a manual of the manufacturer (Carl Zeiss, Oberkochen, Wurtt)

The tables were checked for the usefulness in flow birefringence by WALES

(15) and were found to be very accurate. Compensation carried out in this

way always gives the phase difference except for a multiple of 2T The

mul-tiple IS not directly obtainable in monochromatic light, but is usually

evident if one works with steadily increasing stresses It can also be

de-termined with the aid of white light, as a dark field only occurs in this

case when the phase shift produced by the compensator is equal but opposite

2 4 2 The compensator according to DE SeNARMONT

The compensator of DE S6ARM0NT (14) consists mainly of a quarter-wave

plate adapted to the wave length of the monochromatic light used This

quarter-wave plate is mounted between crossed polarizing prisms so that its

extinction directions coincide with the directions of polarization of the

crossed polarizing prisms The birefringent medium is put between polarizer

and quarter-wave plate in such a way, that the extinction directions of the

birefringent medium make angles of fortyfive degrees with those of the

quarter-wave plate In this way, the axes of the vibration ellipsoid of the

light wave, emerging from the birefringent medium coincide with the

princi-pal directions of the quarter-wave plate It can be shown (16) that -under

these conditions- the beam emerging from the quarter-wave plate is linearly

polarized Its direction of polarization, however, is no longer

perpendicu-lar to the original direction of the analyzer This furnishes the

possibi-lity to extinguish the emerging beam by a suitable rotation of the analyzer

over a certain angle tf)' When the light emerging from the polarizer is

cha-racterized by a vector A of magnitude

A = A sin 2TTVt, (2 14) o

where V is the frequency of the light used, t is the time, and A is the

am-plitude of the wave, then the intensity of the light emerging from the

ana-lyzer IS given by the time average (over one period) of the square of the

projection of the light vector on the direction of the analyzer

1/v

J = \j A^ dt = i A^ sin^Ct)' - 6/2) (2 15) J t o

0

In this equation 6 is the phase difference that arises between the principal

directions of the birefringent medium Therefore, the intensity becomes zero

when

(()' - 6 / 2 = 0 (2 16)

24i' (2.17)

The measurement of the phase difference is reduced to the measurement of an

extinction position of the analyzer.

For small phase differences 6 it is sometimes difficult to determine

the angle ([)' accurately enough. In practice one has to determine the

posi-tion, where the intensity of the light transmitted by the analyzer Is

mini-mal (17, p. 301). Recently, however, methods were developed, furnishing an

important improvement of the determination of extinction positions by the

application of a modulation of the birefringence of the medium, Details

about these methods will be reported in the next section.

2.5 THE MODULATION OF THE BIREFRINGENCE: A TOOL FOR THE ACCURATE

MEASURE-MENT OF EXTINCTION POSITIONS AND PHASE DIFFERENCES

2.5.1 The modulator

In our case, the modulation of the birefringence is obtained by means

of a vibrating glass bar. The bar is supported in the nodal points of the

lowest transversal vibration mode and excited electro magnetically. The

pertinent lowest eigen frequency is close to 1000 Hz. The glass bar has got

a U-shaped cross-section simply by cementing two glass strips B on the edges

of the broader glass strip A (see Fig. 2.4). As a glue Canada balsam has

L

Fig. 2.4 Scheme of the modulator: A ... glass strip, B ... narrow glass strips cemented on A, LL' ... light beam, N ... points of support (in the nodes).

been used. As a consequence of this measure, with bending the neutral stress

free plane is no longer in the middle of the cross-section of A. Complete

compensation of the stretch and compression birefringences, respectively,

which occur on either side of the neutral plane, can no longer take place.

In this way a sinusoidally varying phase difference A of adjustable

A = A sin tot,

o (2.18)

where OJ is the circular frequency of this vibration and A is proportional

to the amplitude of the bar The principle of this modulator has already

been suggested by JANESCHITZ-KRIEGL (17) in his review article. A more

de-tailed description of the indicated system will be given by B. KOEMAN in

due course Various other types of modulators have been described in the

literature (18-22) However, none of them has been proved to be very useful

in practise

2.5.2. Measurement of the extinction angle

In the alignment of polarizer, birefringent medium, analyzer, the

modu-lator IS inserted directly after the birefringent medium. The position of

the principal directions is given m Fig. 2.5. The principal direction I'

of the modulator makes an angle of fortyfive degrees with the polarizer.

Fig. 2.5 Schematic representation of the optical components for the measure-ment of the extinction angle with the aid of modulated light: P,A . . . pola-rizer and analyzer, in crossed posi-tion, 6 ... phase difference of the birefringent medium with principal directions I and II, A . .. phase dif-ference of the modulator with princi-pal directions I' and II'.

The principal direction I of the birefringent medium makes an angle (J" with

the polarizer From the time average (over one period) of the square of the

projection of the light vector on the analyzer, the intensity of the light

emerging from the analyzer can be calculated

J = 1/4 _{A 1-cos 2!J) cos A-sin 2({) cos 6 cos A + s i n 2<i> s i n & s i} 2 1 2 2

n A 1

(2.19)

In this averaging it is implied that the time dependent phase difference A

Expanding eq (2 19) into a Taylor series m powers of A and

disregard-ing terms of higher order than the second, one obtains

2r

o L

_ 2 2 2 2

J = 1/4 A I s i n 2(j)(l-cos 6) + l / 4 ( c o s 2d) + cos 6 s m 2(b) A + _{" 1 o}

2 2 2 1

+ s m 6 s i n 2(1)(A s i n tot) - l / 4 ( c o s 2(j) + cos 6 s i n 2({)) A cos 2ujt

(2 20)

If the intensity is measured with the aid of a photomultiplier and the

signal is investigated on an oscilloscope, one notices a sudden change of

the frequency of the alternating component of the current from to to 2a),

when the angles (() = 0 and (j) = 90 are closely approached The transition

becomes the sharper the smaller A is with respect to 6 With the aid of an o

amplifier tuned to the frequency to, the point of transition of the

fre-quency can be adjusted very accurately

2 5 3 Measurement of the birefringence accordtng to DE SeNARMONT

In the alignment according to DE S6NARM0NT (14), vz polarizer,

bire-fringent medium, quarter wave plate and analyzer, the modulator is inserted

directly after the birefringent medium The various optical components are

set with respect to each other as schematically given in Fig 2 6 The

direction of transmission of the polarizer P coincides with the suitable

principal direction of the quarter wave plate The principal directions of

the birefringent medium I and of the modulator I',make both an angle of

fortyfive degrees with the transmission direction of the polarizer The

ex-tinction position of the analyzer is found at an angle ())' from the crossed

position With the polarizer In the same manner, as shown in the previous

Fig. 2.6 Schematic representation of the optical components for the measurement of the flow birefringence with the aid of modulated light: P . . . polarizer, A . . . analyzer, %X ... quarter-^ave plate,

6 ... phase difference of the birefrin-gent medium, A ... phase difference of the modulator.

section, an expression can be derived for the intensity J of the light

emerging from the analyzer:

J = i A^ sin2(())' - 6/2 + 4 A ) (2 21)

Expanding this equation into a Taylor series with respect to A, neglecting 2

terms of higher order than A , one obtains

2 r 2

J = 1/4 A 1 - c o s ic + 1/4 c o s K A + s m K A s i n tot + o L o o

+ 1 / 4 c o s K A c o s 2ojt ( 2 . 2 2 )

w i t h K = 4 ) ' - 6 / 2 .

As with the measurement of the extinction angle, we notice that there is a

point of transition in the frequency from to to 2to when K = 0 is closely

approached. The following expression is found for the measured phase

dif-ference 6:

6 = 24)' (2.23)

Thus, the measurement of the phase difference is reduced to that of an

2.6 REFERENCES

1) LODGE, A.S., Elastic Liquids, p. 199, London-New York, Academic Press (1964).

2) OKA, S., in F.R. Eirich ed., Rheology III, Academic Press, New York (1960).

3) FREEMAN, S.M. and K. WEISSENBERG, Nature 161, 324 (1948). 4) MEISSNER, J., J. Appl. Polym. Sci., 16, 2877 (1972). 5) BAGLEY, E.B., J. Appl. Phys. 28, 624 (1957).

6) see e.g. BRYDSON, J.A., Flow Properties of Polymer Melts, Butterworth and Co Ltd, London (1970).

7) MORRISON, T.E., L.J. ZAPAS and J.W. DE WITT, Rev. Sci. Instr. 2j6, 357 (1955).

8) DEKKING, P., Determination of dynamic mechanical properties of high polymers. Doctoral Thesis, Leiden 1961. Leiden: Luctor et Emerge. 9) DUISER, J.A., Het Visco-Elastice Gedrag van Twee Polycarbonzuren in

Water, Doctoral Thesis Leiden, 1965. Leiden: Druco N.V.

10) DEN OTTER, J.L., Dynamic Properties of Some Polymeric Systems. Doctoral Thesis Leiden, 1967. Leiden: Druco N.V.

11) FERRY, J.D., Visco-Elastic properties of polymers, p. 88. New York-London: Interscience 1961.

12) LAMB, J. and P. LINDON, J. Acoust. Soc. Am. 41, 1032 (1967). 13) EHRINGHAUS, A., Z. Krist. 76, 315 (1931).

14) DE S6NARM0NT, H., Ann. Chim. Phys. (2), Bd. 73, 337 (1840).

15) WALES, J.L.S., "The Application of Flow Birefringence to Rheological Studies of Polymer Melts", Monograph, Delft University Press (1976). 16) WALKER, M.J., Am. J. Phys. 22, 170 (1964).

17) JANESCHITZ-KRIEGL, H., Adv. Polym. Sci., 6, 170 (1969). 18) WAYLAND, H., Compt. Rend. 249, 1228 (1959).

19) WAYLAND, H., and J. BADOZ, Compt. Rend. 250, 688 (1960). 20) LERAY, J., and G. SCHEIBLING, Compt. Rend. 251, 349 (1960). 21) LERAY, J., and PH. GRAMAIN, J. Chim. Phys. 60, 1396 (1963).

CHAPTER 3

A RE-DESIGNED CONE-AND-PLATE APPARATUS FOR

THE MEASUREMENT OF THE FLOW BIREFRINGENCE

*)

OF POLYMER MELTS

3 .1 INTRODUCTION

Flow birefringence may be considered as one of the more productive

ex-perimental methods suitable for the investigation of the rheological

pro-perties of polymeric systems. The main advantage of this method lies in the

possibility to obtain a rather accurate knowledge of the state of stress in

a flowing polymer, without using a mechanical measuring device.

The operational principles of flow birefringence were extensively

des-cribed by one of the present authors m his review article (1) We are

in-terested m the relationship between the optical and mechanical properties

of the melt. These properties may be described in terms of the refractive

index and the stress ellipsoids. The link between these two tensors is

formed by the so-called "stress-optical law", which claims the

proportion-ality between the deviatoric tensor components (i e An = C.Ap). It also

includes the coaxiality of the mentioned ellipsoids.

As far as solutions were concerned, the validity of the stress-optical

law has been substantiated for steady shear flow over wide ranges of shear

rates and concentrations, and for various types of polymers by a number of

authors (1-6). The validity of the stress-optical law, however, has been

questioned for transient flow by JANESCHITZ-KRIEGL (1), WAYLAND (7) and

HARRIS (8) but no experimental results were presented. The first attempt to

check the stress-optical law experimentally for the transient region in slow

shear flow of a polymer melt was made by JANESCHITZ-KRIEGL and GORTEMAKER

(9). Additionally, only few experimental results are available for steady

shear flow of polymer melts at high shear rates (10,11).

In this Chapter, a cone-and-plate apparatus will be described. This

apparatus was originally designed to furnish information on steady shear

*^ GORTEMAKER, F.H., M.G. HANSEN, B. DE CINDIO, H. JANESCHITZ-KRIEGL,

flow, in a range of shear rates higher than usually accessible in a

cone-and-plate apparatus At the same time, however, this apparatus happened to

give reasonably accurate results also in the low shear rate range for

fluids responding rather slowly to a step in the shear rate

Some typical flow birefringence measurements will be presented for

three commercial polymer melts over a wide range of shear rates In order

to check the stress-optical law, use is made of additional oscillatoric

me-chanical and capillary viscometer measurements

Also some typical measurements in the transient region of slow flow

are presented to show the rather wide applicability of the apparatus In

this connection some results of measurements on a Weissenberg rheogoniometer,

as kindly supplied by Drs LAUN and HuNSTEDT (BASF Ludwigshafen), are also

incorporated

Numerical analysis of the flow was performed to obtain an understanding

of the actual velocity profiles for different nominal shear rates In this

way also an estimate could be made of the influence of frictional heat on

the experimental results

3 2 DESCRIPTION OF THE APPARATUS

The experimental set-up consists of three different main parts, which

will be described separately the cone-and-plate apparatus containing the

test section, the optical system and the driving system

The operational principle of the cone-and-plate apparatus is

illu-strated in Fig 3 1 As already pointed out, the apparatus is similar to

the one described by WALES and JANESCHITZ-KRIEGL (12) In both designs there

are no free surfaces of the sample The test section pos 7 is an annulus

bounded by the rotating plate pos 6, the stationary cone pos 5 and the

stationary inner and outer cylindrical surfaces These surfaces are formed

by an inner cylindrical part located in the centre of the ring and by an

outer cylindrical part on which the heating elements are arranged It

ap-pears that the complete enclosure of the sample achieved m this way

effect-ively reduces degradation, which evidently occurs, if the rim of the sample

is in contact with air The test section is filled by injecting molten

po-lymer through a hole pos 9 made in the stationary cone As the popo-lymer is

forced into the apparatus, the air contained in the gap can apparently

Fig. 3.1 Cross-section through heart of cone-and-plate system:

(1) linearly polarized light beam, (2) reflection prism, (3) inner window,

(4) outer window, (5) stationary cone, (6) rotating plate, (?) test

section, (8) blind hole for thermocouple, (9) sample injection hole,

(10) elliptically polarized light beam, (11) analyzer.

(optical glass BK 7, Schott and Gen. (Borfe)) pos. 3 and pos. 4 are located

along a major diameter (B-B) in the cylindrical bounding surfaces of the

test section. This type of glass has a low thermal cubic expansion

coeffi-cient (2.3 * 10 C ) and a low residuel birefringence.

This apparatus represents a further development of the previously

men-tioned one (12) being essentially different only with respect to the

propa-gation of light. In fact, with respect to the earlier apparatus, the

direc-tion of the light beam is reversed, following a suggesdirec-tion by WALES. The

explanation for this change will be given below. In the present arrangement,

the light enters from the right along the axis A-A, and is reflected by

means of a glass prism pos. 2 which is located in the centre of the inner

cylindrical part. This prism reflects the light radially outwards along the

Fig. 3.2 Schematic drawing of the optical measurement system:

(1) high pressure mercury lamp, (2) lens, (3) aperture, (4) interference filter, green 546.1 nm, (5) lens, (6) polarizer, (7) reflection prism,

(8) inner window, (9) polymer sample, (10) outer window, (11) Ehringhaus compensator, (12) analyzer, (13) photomultiplier, (14) high speed (ultra-violet light beam) recorder.

A scheme of the optical system is presented in Fig. 3.2. The light

source pos 1 is a high pressure mercury lamp. Optical elements along the

line A-A are positioned to focus the light beam through the test section.

Linearly polarized light is obtained by means of a polarizing sheet

(POLA-ROID H N 22 X 0.35") pos 6. An analyzer pos 12 of the same type as the

polarizer is located after the upper window. Between the upper window and

the analyzer, there is a holder pos. 11 for a compensator which enables the

measurement of the birefringence. A filter pos. 4 (Interference filter

Fil-traflex-B-10, manufactured by Balzers with a wavelength of 546.1 mm and a

tolerance of ^^ 0.15%) is inserted into the beam to produce monochromatic

The system built in this way is a typical linear polariscope (13) with

crossed polarizer and analyzer (dark field). The optical elements from pos.

1 to 6 of the figure are mounted on an optical bench which hinges on the

line B-B. Also the prism, the two windows and the analyzer are rotated

around the same line B-B, being rigidly connected to the hinged optical

bench Hence, the vertically (or horizontally) polarized incident beam

al-ways remains polarized parallel (or perpendicular) to the incidence plane

of the reflection prism. Therefore, the state of linear polarization of the

incident beam remains uneffected by the prism . A vernier scale permits

the reading of the angle of rotation of the optical bench around B-B In

this way the determination of the position of the polarization direction

with respect to the flow field is achieved.

When the sample is deformed by the flow, it becomes optically

aniso-tropic and shows extinction positions and a certain amount of birefringence.

Usually, the extinction angle X is defined as the angle smaller than

forty-five degrees, occurring between one of the extinction positions and the

direction of the streamlines To determine Xi the direction of polarization

is rotated until it is aligned with the corresponding axis of the

refrac-tive index ellipsoid. This is achieved by rotating the optical bench around

the line B-B until the intensity of the light beam emerging from the

ana-lyzer IS minimized. Then a reading is made from the vernier scale. The

in-determinateness of the zero-position of the vernier scale is eliminated by

making another reading for the reversed flow direction. Half of the

dif-ference between these readings gives the extinction angle x The

zero-position is obtained by averaging these values. The birefringence An is

measured after the insertion of the EHRINGHAUS compensator (manufactured

by CARL ZEISS) in the previously mentioned holder and insertion of

mono-chromatic filter at pos. 4, Fig. 3.2. The principle of operation of this

type of compensator is described m the literature (14).

In the original version (12), corrections were necessary for the change

which the elliptically polarized light emerging from the test section,

nrm

^

Ir r

O

iS'fO

Of-2 3 6 9 10

(1) frequency synthesizer, (2) synchronous motor, (3) gear box, (4)

chopped light tachometer, (5) electronic counter, (6) electrical magnetic clutch, (7) clutch control clock, (8) high speed (ultra-violet light beam) recorder, (9) gearboxes, (10/ rotating plate coupling.

The scheme of the driving system is depicted m Fig. 3.3. It consists

of a synchronous motor, which is fed by a frequency synthesizer in order

to supply a stepless speed control over one decade of frequency. A wide

range of shear rates is obtained with the aid of an assemblage of

inter-changeable speed reduction gear-boxes. The rotational speed is measured by

a chopped-light tachometer connected to an electronic counter. The

constan-cy of the drive is within 0.1%. The rotating plate of the test section is

connected (through one of the gear-boxes) to the driving system by means

of an electrical clutch The engagement and disengagement of the clutch is

controlled by a clock The response of the clutch has been determined with

the empty test section It was measured that full speed of the rotor is

achieved within 10 ms

This drive system enables the investigation of material transient

response to a step- and boxlike function of shear rate. Thus, stress growth

after a period of rest, stress relaxation after a certain amount of shear

and stress relaxation after steady shear flow can be investigated. For these

ultra-violet light recorder as shown in Fig. 3.3

At this point it should be admitted that the development of high speed

recording is not yet finished. The change of extinction angle with time is

readily measured by recording the times at which the isocline passes

seve-ral presettings of the crossed polarizers with the photomultiplier tube

pos. 13 and ultra-violet light recorder pos. 14 of Fig. 3.2. However, the

handling of the EHRINGHAUS compensator is much too time consuming for

ra-pidly responding materials. In order to improve recording speed and

accu-racy the following two modifications were made

1 ) modulation of the birefringence with the aid of a rotating mica

plate, in order to measure weak but quickly changing

birefringen-ces .

11) the utilization of circularly polarized light in order to avoid

interference of isoclines and fringences and thus to enable

un-disturbed counting of quickly passing fringes, when the

bire-fringence becomes large

Finally, we shall make some comments on the temperature control.

Se-veral heating elements are positioned around the test section. A check for

the absence of any temperature gradient in the gap was made in the empty

test section Two thermocouples were used (see Fig. 3.1) one was inserted

into the test section through the injection hole (pos 9 ) , while the other

one was located at the bottom of a blind hole (pos 8) in the stationary

cone, near the windows. The distance from the cone surface to the bottom

of the hole is 1 mm. In this way we were able to measure the temperature

in two diametrically opposite points of the gap The temperature differen-o

ce between these two points was less than 0.2 C. This confirms that no

appreciable temperature gradients are present in the gap due to the

non-uniform distribution of the heating elements. The temperature during the

test IS measured by means of a thermocouple inserted in the blind hole

3.3 MATERIALS AND RHEOLOGICAL CHARACTERIZATION

Three different commercial polymers have been used, a high density

polyethylene (Manolene 6050), a low density polyethylene (sample A (22-24))

and a polystyrene (Hostyren N 4000 V ) .

cylindrical heated mould and melted under vacuum. Then the material was

compressed and cooled under pressure. The value of the pressure was about 7 -2

1.5 * 10 N m . I n this way a cylindrical bar (with a diameter of 2 cm

and a lenth of 15 cm) is obtained. This bar is mechanically machined to

obtain the right shape and size of the injected sample.

In order to check the kinematics of our system and the validity of

the stress-optical law, a rheological characterization of the polymers was

necessary. For this purpose, the steady shear flow curves over a rather

wide range of shear rates were measured. This was achieved with the aid of

different mechanical tests. Capillary viscometry was applied to obtain the

flow curves in the high shear rate region. Dynamic mechanical measurements

were used to enable extrapolation of viscosity-shear rate curves to low

shear rates, where capillary viscometry fails

Values of steady state viscosity at different shear rates were

deter-mined using a capillary viscometer designed by J. VAN LEEUWEN and

R. VAN DER VIJGH at the Plastic and Rubber Institute T.N.O. (Delft). The

data were corrected according to BAGLEY (15) for entrance effects. The

shear rate q was evaluated at the wall according to RABINOWITSCH (16). The

covered range of shear rates is from I s to about 1000 s

Values of the storage and loss moduli (G' and G") at rather low

cir-cular frequencies (o, were obtained with the aid of an automatic dynamic

viscometer, designed by K. TE NIJENHUIS at this laboratory. This apparatus

IS an improved version of DEN OTTER's dynamic viscometer (17,18). It allows

the obtaining of accurate data also in the range of very low frequencies

where the phase angle differs only little from ninety degrees. Details of

this apparatus will be published in due course. The values of G' and G"

were obtained as functions of frequency and temperature. Use was made of

the time-temperature superposition principle (19) in order to obtain

mas-ter curves over an extended range of frequencies at the chosen reference o

temperature (179 C for the high density polyethylene).

An empirical relation, proposed by COX and MERZ (20), was used to

ob-tain the steady shear viscosity ri as function of the shear rate q from the

reduced values of G' and G". This relation reads:

n(q) = I n * (to) I _ (3.1)