EUROPEAN

POLYMERJOURNAL

European Polymer Journal 41 (2005) 1484–1492

www.elsevier.com/locate/europolj

Validation of a model to predict birefringencein injection molding

Roberto Pantani *

Department of Chemical and Food Engineering, University of Salerno, I84084 Fisciano (SA), Italy

Received 21 January 2005; received in revised form 3 February 2005; accepted 7 February 2005

Available online 16 March 2005

Abstract

The current goal in the simulation of injection molding is the description of material morphology. The path to reach

this goal passes through the prediction of molecular orientation and strain, namely the molecular conformation. To

obtain this information, the viscoelastic nature of the polymer must be taken into account. The aim of this paper is

to check if a simple, recently proposed, non-linear dumbbell model is sufficiently accurate to quantitatively describe

birefringence distribution in injection molded PS samples. To this goal, a series of rheological measurements were per-

formed in a parallel plate rheometer, measuring in the meantime the birefringence. By choosing an appropriate stress-

optical coefficient, the model could describe the whole set of data. The results obtained allowed to reinterpret some

results of molecular orientation in injection molding and to reach a quantitative description of data of birefringence

distribution in molded PS samples.

� 2005 Elsevier Ltd. All rights reserved.

Keywords: Polystyrene; Flow birefringence; Dumbbell model; Molecular orientation; Injection molding

1. Introduction

The modeling of molecular conformation during flow

in polymer melts is assuming an increasing importance

in the simulation of polymer processing in general and

of injection molding in particular [1–6]. Indeed, if in

the past process simulations were mainly aimed at the

solution of balance equations to describe flow rates,

temperature distributions and pressure histories, the cur-

rent goal is the description of material morphology. The

path to reach this goal passes through the prediction of

0014-3057/$ - see front matter � 2005 Elsevier Ltd. All rights reserv

doi:10.1016/j.eurpolymj.2005.02.006

* Tel.: +39 089964141; fax: +39 089964057.

E-mail address: rpantani@unisa.it

molecular orientation (which induces anisotropy in

material properties) and strain (which leads to specific

characteristics, such for instance stiffness), namely the

molecular conformation. To obtain this information,

the viscoelastic nature of the polymer must be taken into

account. Among viscoelastic models, those ones which

are based on molecular models provide obviously a di-

rect answer to the problem, being able to describe molec-

ular deformation.

The desirable characteristics of a suitable model can

be briefly summarized as:

• It should be simple and easily implementable in a

software for polymer processing. The level of detail

should of course be compared with the physical

ed.

Table 1

Values adopted for parameters of Eqs. (1) and (2) [8]

A1 3.77

A2 181.06 K

g* 1856 Pa s

D2 493 K

N 0.252

s* 30,800 Pa

R. Pantani / European Polymer Journal 41 (2005) 1484–1492 1485

parameters one wants to describe. To this regard,

molecular strain and orientation should be consid-

ered as a minimum target.

• It must be consistent with material rheology. This

(apparently obvious) point deserves a brief clarifica-

tion. A ‘‘decoupled method’’ is commonly adopted

in the simulation of polymer processes [4–6]. This

method relies on the assumption that the elastic

behavior of the polymer only marginally influences

the flow kinematics. The melt is considered as a

non-Newtonian viscous fluid (whose viscosity is

described for instance by a Cross or Carreau model)

as for the description of pressure and velocity gradi-

ents. The obtained kinematics is used to describe the

evolution of molecular orientation by means of the

viscoelastic model. Obviously, the viscous and visco-

elastic models must provide the same results when

applied in the same steady-state conditions.

• It should contain the minimum possible number of

parameters, whose values should be easily attained

by means of a standard characterization.

A standard and powerful mean to obtain informa-

tion about orientation in polymer samples is to ana-

lyze the distribution of birefringence. For some

materials, like for instance polystyrene, birefringence

is in fact directly correlated to the frozen-in molecular

orientation. It is worth mentioning that the prediction

of birefringence is itself a desirable property to pre-

dict, since for some parts for optical applications the

control of birefringence distribution is crucial. The

ability of a model to predict birefringence is therefore

a quite valuable feature.

The aim of this paper is to check if a recently pro-

posed non-linear dumbbell model corresponds to the

characteristics listed above. To this goal, a series of rhe-

ological measurements were performed in a parallel

plate rheometer, measuring in the meantime the birefrin-

gence (which for polystyrene is an index of material con-

formation). The whole set of data was then analyzed by

means of the non-linear dumbbell model and a proper

choice of the stress-optical coefficient was made. On

the basis of the results obtained, it was possible to quan-

titatively describe the distribution of birefringence in

injection molded samples.

2. Experimental

2.1. Material

An atactic polystyrene (Dow PS 678E) was used in

this work. The rheological behavior of polymer melt

was characterized in the literature for the temperature

range 473–503 K [5,7,8] and was well described by a sim-

ple cross model:

gðc0; T Þ ¼ g�aðT Þ

1þ g�aðT Þc0s�

� �1�n ; ð1Þ

where the thermal shift factor, a(T), is given by the WLF

equation

aðT Þ ¼ 10�A1 ðT�D2ÞA2þT�D2

� �: ð2Þ

The constants for this model are given in Table 1.

2.2. Rheological and optical measurements

The device used in this work is an ARES (Rheomet-

rics) rheometer, equipped with an Optical Analysis

Modulus OAMII. A parallel-plates configuration was

adopted, by using two quartz disks of radius R =

19.1 mm. The incident light beam passed through the

sample at a radius RB = 15.5 mm.

Rheo-optical tests were conducted at a temperature

of 463 K in rate-sweep mode, i.e. increasing the shear

rate in consecutive steps and recording at each step the

plateau values for torque (T), normal force (Nf) and

birefringence (Dn). The rheological measurements were

found to be reliable up to values of shear rate (evaluated

at r = R) of about 2 s�1. For higher values some material

was pulled out from the edge of the plates thus making

the measurements unreliable. In spite of this, data of

birefringence could still be considered reliable, being

the material still present at r = RB.

The complete set of data is reported in Table 2. Since

the laser beam crossed the sample at r = RB, two shear

rates are reported in the table, considering a linear

dependence of shear rate upon radius.

The measures of torque and normal forces allow the

evaluation of the shear stress, srh, and of the difference

between the first and the second normal stress differ-

ences, N1 � N2, at r = R according to the equations [9]

srhðRÞ ¼T

2pR33þ d lnðT Þ

d lnðc0RÞ

� �; ð3Þ

N 1ðRÞ � N 2ðRÞ ¼Nf

pR22þ d lnðNfÞ

d lnðc0RÞ

� �: ð4Þ

Results are reported in Fig. 1.

Fig. 1. Measurements of the shear stress, srh, and of the

difference between the first and the second normal stress

differences, N1 � N2, for different shear rates.

Table 2

Measurements performed during experiments: parallel plate

device, T = 190 �C, R = 19.1 mm (birefringence data are mea-

sured at RB = 15.5 mm)

Shear rate

at R, c0R[s�1]

Shear rate

at RB, c0B[s�1]

Normal

force, Nf

[N]

Torque,

T

[N m]

Birefringence

10�6

(±20%)

0.053 0.043 0.10 0.006 0.70

0.095 0.077 0.10 0.010 1.49

0.126 0.103 0.10 0.013 1.67

0.169 0.137 0.18 0.018 1.34

0.225 0.183 0.24 0.024 2.37

0.300 0.244 0.25 0.031 1.85

0.400 0.325 0.61 0.038 4.85

0.492 0.400 1.29 0.045 10.00

0.533 0.434 1.34 0.045 7.35

0.711 0.579 1.87 0.055 11.50

0.738 0.600 2.21 0.055 18.00

0.930 0.757 2.95 0.070 17.00

1.230 1.001 3.41 0.092 25.00

1.850 1.505 5.80 0.113 42.00

2.450 1.993 7.29 0.167 53.00

3.000 2.441 62.00

4.000 3.255 85.00

5.000 4.068 93.00

1486 R. Pantani / European Polymer Journal 41 (2005) 1484–1492

3. Discussion

3.1. The non-linear dumbbell model

In this work, a non-linear formulation of the elastic

dumbbell model [4] was adopted to describe the visco-

elastic nature of the polymer. If R is the end-to-end

vector of a molecular chain, and the symbol h i denotesthe average over the configuration space, it is possible

to define the fractional ‘‘deformation’’ of the popula-

tion of dumbbells with respect to the equilibrium con-

formation as

A ¼ 3

hR20i½hRRi � hRRi0�; ð5Þ

where hRRi is the second-order conformation tensor,

hRRi0 is the value of hRRi at rest, when the end-to-

end distance of the molecular chain is hR20i ¼ trhRRi0.

According to this definition, the final constitutive

equation for a dumbbell population can be written as

[10]:

D

DtA�rvT � A� A�rv ¼ � 1

kAþrvþrvT ; ð6Þ

where $v is the velocity gradient and k is the relaxation

time.

The polymer contribution to the stress tensor is

obtained from the polymer conformation as

s ¼ GA; ð7Þ

where G is the modulus of the polymer.

In the elastic dumbbell model the relaxation time, k,is not considered to be dependent on shear rate, and it is

well known that with this choice the model is not able to

predict the shear thinning behaviour of polymer melts. If

however k and G are allowed to vary with shear rate and

temperature, the consistency between material steady-

state shear viscosity and Eqs. (6) and (7) can be

preserved. In particular, the following functions are

suggested by Pantani et al. [4]:

kðT ; c0Þ ¼ k�aðT Þ

1þ k�aðT Þc0k

h i1�m ; ð8Þ

GðT ; c0Þ ¼ gðT ; c0ÞkðT ; c0Þ ; ð9Þ

where g and a are the shear viscosity and the thermal

shift factor already introduced in Eqs. (1) and (2). By

introducing Eqs. (7)–(9), the model depicted above can

be considered as a modified version of the White–Metz-

ner model: in order to describe the relaxation time

dependence on shear rate, a cross-WLF equation (Eq.

(8)) was found in reference 4 to be suitable to describe

data of relaxation time obtained by measurements

mainly in dynamic mode. The thermal shift factor for

the relaxation time was experimentally found [4] to be

equal to the thermal shift factor for viscosity. Eq. (9)

essentially assures that the shear viscosity of the material

is well described by the dumbbell model. It is worth

mentioning that, according to the model depicted above,

material viscoelastic behavior is described with the use

of only one relaxation time. However, since relaxation

time is a function of shear rate, it essentially describes

a series of infinite relaxation modes, each one playing

a role at a given shear rate.

The parameters to use in Eq. (8) were identified by

Pantani et al. [4] for the same material adopted in this

Table 3

Values adopted for parameters in Eq. (8) [4]

k* 0.315 s

m 0.21

k 0.354

Fig. 2. Symbols: relaxation time, k, and modulus, G, as derived

from experimental data reported in Fig. 1 analyzed according to

Eqs. (12) and (13). Lines: description of the same material

functions according to Eqs. (8) and (9), with parameters listed

in Tables 1 and 3.

R. Pantani / European Polymer Journal 41 (2005) 1484–1492 1487

work and are listed in Table 3. These parameters will be

left unchanged in this work and adopted to compare the

model predictions with the new experimental results

obtained in this work.

The advantage of adopting the model depicted above

relies on its simplicity: the knowledge of the two relevant

material parameters (namely the relaxation time and the

modulus, which do not need any discrete spectrum) al-

lows by means of Eq. (6) the calculation of the polymer

conformation once the velocity field is known. The stress

tensor is then directly calculated from the conformation

itself. The model is obviously expected to provide reli-

able results in shear-dominated flow fields (like those

taking place in injection molding).

3.2. Analysis of rheological data by the dumbbell model

The parallel plate rheometer realizes a steady-state

shear flow in which c0 ¼ c0Rr=R, vh = c 0z, vr = 0, vz = 0.

For this flow field, the only non-zero components of

the deformation tensor A are calculated from Eq. (6) as

Ahh ¼ 2ðkc0Þ2;Ahz ¼ Azh ¼ kc0: ð10Þ

According to Eq. (7), there are therefore only three

components of the stress tensor, namely

shh ¼ 2Gðkc0Þ2;shz ¼ szh ¼ Gkc0 ð11Þ

and thus, being the second normal stress difference equal

to zero, one obtains for this model

N 1 � N 2 ¼ N 1 ¼ shh: ð12Þ

The measurements of shz and of (N1 � N2) during a

test with a parallel plate rheometer allow thus the direct

determination of the modulus G and of the relaxation

time k for the dumbbell model, according to the

equations

k ¼ N 1

2shzc0ð13Þ

and

G ¼ shzkc0

ð14Þ

which are obtained by a simple rearrangement of Eqs.

(11) and (12).

The values of k and G calculated from the data re-

ported in the experimental section according to Eqs.

(13) and (14) are reported in Fig. 2, together with the re-

sults of Eqs. (8) and (9) (with parameters taken from

Ref. [4]). The comparison is satisfactorily, thus showing

that the model depicted above is able to describe mate-

rial rheology in steady-state shear flow. On the other

hand, the procedure followed to obtain Fig. 2 confirms

that the parameters of the model are easily obtained

from a standard material characterization.

3.3. Analysis of birefringence by the dumbbell model

According to Eq. (5), the normalized (with respect to

its rest value) conformation tensor can be written as

hRRihRRi0

¼ Aþ 1: ð15Þ

It is useful in the following to refer to a local frame of

reference x(parallel to r)–y(parallel to h)–z, located at

a distance r from the rotation axis of the rheometer.

In this frame, the tensor A + 1 can be expressed as

(after Eq. (10))

Aþ 1 ¼1 0 0

0 1þ 2c02k2 c0k

0 c0k 1

0B@

1CA ð16Þ

which defines an ellipsoid describing the deformation of

the dumbbells� population subjected to a steady-state

shear flow.

The eigenvectors of the tensor A + 1 define a new

frame of reference x*–y*–z*. In the particular case of

the rheological tests analysed in this work (as depicted

in Fig. 3) x* is parallel to x, the plane y*–z* is parallel

to the plane y–z and a is the angle between the axes y*

and y (or between the axes z* and z). In this new frame

of reference, A + 1 is represented by a diagonal matrix,

in which each value represents the strain of the dumb-

bells� population along a principal direction, and thus

Fig. 3. Local frame of reference adopted to analyse the polymer

conformation tensor. x = vorticity direction, parallel to r;

y = flow direction, parallel to h; z = velocity gradient direction;

x*, y* and z* are the axes of the polymer frame of reference.

1488 R. Pantani / European Polymer Journal 41 (2005) 1484–1492

the values on the diagonal identify the local anisotropy

of the polymer chain population.

At constant temperature the relaxation time k is func-tion of shear rate only, through Eq. (8), and thus in fact

the shape of the ellipsoid defined by Eq. (16) is deter-

Fig. 4. Deformation of the dumbbells� population at T = 190 �C and

Fig. 3: x*, y* and z* are defined by the major axes of the ellipsoid.

mined by the value of c 0 alone. The deformation of the

dumbbells� population as described by Eq. (16) is re-

ported in Fig. 4 for T = 190 �C and for increasing shear

rate: At low shear rates, neither particular orientation

nor anisotropy is present and Eq. (16) only slightly de-

parts from a sphere of unit radius; as shear rate in-

creases, the dumbbells� population becomes more

stretched in the y (=flow) direction, the angle between

y* and y, namely the angle of orientation, decreases

and a marked anisotropy sets in for shear rates higher

than about 1 s�1.

The macroscopic refractive index of a polymer

chain is different in the directions parallel to and nor-

mal to the chain main axis [11]. Birefringence mea-

surements have been extensively used to examine

steady and transient flows of polymeric fluids [12–

16], and in particular to determine the local anisot-

ropy in the conformation of the polymer chains. The

resulting expression for the refractive index tensor n

can be written as

n ¼ CGðAþ 1Þ þ const1; ð17Þ

on increasing shear rates. The reference frames are defined in

R. Pantani / European Polymer Journal 41 (2005) 1484–1492 1489

where C is the stress-optical coefficient and G the char-

acteristic elastic modulus of the fluid. The birefringence

is the difference in the principal values of the projection

of the refractive index tensor n onto a plane perpendic-

ular to the light path.

A polarized light beam traveling along the x direction

(that is the vorticity direction) would analyze a projec-

tion of the refractive index onto the y�z plane.

The measured birefringence would thus be given by

Dy�z ¼ n�yy � n�zz ¼ CGðA�yy � A�

zzÞ

¼ CG2c0kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ c02k2

q: ð18Þ

The laser beam used in this work travels along the

z direction and thus it analyses a projection of the

refractive index onto the x�y plane. The birefringence

measured by the device adopted in this work is given

by

Fig. 5. Projections onto the planes perpendicular to the x (in black) an

conformation.

Dx�y ¼ nxx � nyy ¼ CGðAxx � AyyÞ ¼ CG2c02k2: ð19Þ

As clear from Fig. 5, a configuration in which the laser

beam travels along the vorticity axis provides a more

complete information with respect to the configuration

adopted in this work. It would, however, require the

use of a couette device rather than a parallel plate cell.

The experimental data of birefringence can be

adopted to define a value of the stress-optical coefficient

in Eq. (19). The value which allowed the best description

of data is C = 5 · 10�9 Pa, which is perfectly in line with

literature values [17,18].

The comparison between model predictions and

experimental determinations is reported in Fig. 6. The

good agreement is a confirmation that the dumbbell

model described in this work can be successfully em-

ployed to describe the local anisotropy in the conforma-

tion of the polymer chains and, adopting the correct

d z directions (in grey) of the ellipsoid representing the polymer

Fig. 6. (a) Comparison between experimental measurements

of birefringence (filled squares) and the model predictions

(dotted line). Predictions obtained considering the vorticity

(y–z) plane are reported as a solid line. (b) Same results, on a

log–log scale.

1490 R. Pantani / European Polymer Journal 41 (2005) 1484–1492

value for the stress-optical coefficient, to predict

birefringence.

Fig. 7. Comparison between experimental measurements of birefrin

configuration) and the predictions obtained by assuming the validity

3.4. A quantitative prediction of birefringence

distribution in injection moldings

In a previously published paper [4] some experimen-

tal results of birefringence distribution in injection mold-

ing samples made of the same polystyrene adopted in

this work were presented. The cavity was a simple rect-

angular slab (length 120 mm, width 30 mm, thickness

2 mm), and two gate thickness were adopted: 1.5 mm

(thick gate configuration), and 0.5 mm (thin gate config-

uration). For both configurations, the holding pressure

was 450 bar, the holding time 15 s, the injection temper-

ature was 240 �C and the mold temperature 30 �C. Inthat work, injection molding tests were simulated by a

software code in which the same model considered

above was implemented adopting a decoupled approach,

as described in the introduction. The experimental distri-

bution of birefringence inside the samples were then

compared with an ‘‘orientation index’’, namely the max-

imum eigenvalue of the tensor A, which resulted to qual-

itatively describe the data. The further step made in this

work was to run again the simulations by assuming the

validity of the stress-optical rule (Eq. (17)) with the value

of the stress-optical coefficient identified above. At each

distance from the mold wall the value of birefringence

was calculated at solidification time, solidification being

assumed to take place when the relaxation time is one

order of magnitude higher than the time needed to reach

the glass transition temperature (Tg = 373 K for PS).

The comparison between experimental data and simula-

tion results at several positions along the flow-path is re-

ported in Fig. 7 (thick gate configuration) and in Fig. 8

(thin gate configuration). It is clear that the model can

quantitatively describe the data in all positions and for

both configurations.

gence distribution in injection molded samples [4] (thick gate

of the stress-optical rule.

Fig. 8. Comparison between experimental measurements of birefringence distribution in injection molded samples [4] (thin gate

configuration) and the predictions obtained by assuming the validity of the stress-optical rule.

R. Pantani / European Polymer Journal 41 (2005) 1484–1492 1491

4. Conclusions

In this work, birefringence data were collected during

rheological measurements performed on an amorphous

polystyrene in a parallel plate rheometer.

A simple non-linear dumbbell model was used to de-

scribe polymer conformation during flow. The non-linear-

ity of the dumbbells was accounted for by allowing the

two material parameters, namely the relaxation time and

themodulus, to be function of temperature and shear rate.

The consistency between the predictions of the model

and the rheological measurements performed in this

work was checked by comparing the evolution of relax-

ation time and modulus with shear rate.

The predictions of the model as for the conformation

were also successfully compared with the birefringence

results, by assuming a proper value of the stress-optical

coefficient.

The results obtained allowed to reinterpret some re-

sults of molecular orientation in injection molding and

to reach a quantitative description of birefringence dis-

tribution in injection molded PS samples.

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