The Role of the Interaction Coefficient

in the Prediction of the Fiber Orientation

in Planar Injection Moldings
A. J. PONTES, N. M. NEWS, andA. S. POUZADA

Department of Polymer Engineering

Universidade do Minho
Campus de Azuri.m
4800-058GuimarEes,Portugal

The mechanical properties of injection molded parts in glass reinforced materials

are sensitive to processing. A successful design requires a good estimate of the
product performance before production. Its performance is strongly affected by the
fiber orientation field set up during processing. The fiber orientation pattern is
complex and varies three-dimensionally in the moldings. Some commercial simula-
tion programs already allow the prediction of the fiber orientation induced during
the flow by the associated stress field. The results from the simulations are depend-
ent on a parameter accounting for the interactions between fibers during the flow,
known as the fiber interaction coefficient. In this paper the effectiveness of the in-
teraction parameter on controlling the predicted patterns of the fiber orientation is
studied. This is done by comparing and analyzing the experimental data and the
corresponding predictions.

1. INTRODUCTION for the fiber-fiber interactions during flow. More re-

cently Fan et al. (6) developed a promising direct nu-
pon injection molding of fiber reinforced thermo-
U plastic, complex patterns of fiber orientation re-
sult through the thickness and along the flow path of
merical simulation of fiber-fiber interactions in shear
flow but requiring too much computation time.
No complete theory or model is yet available to fully
the moldings (1). Most of the properties of the com- predict the fiber orientation resulting from the injection
posite depend on the distribution of fiber orientation. molding process. Some existing models allow the calcu-
Thus, a precise determination of the composite me- lation of the rheological and mechanical properties of
chanical properties requires either the prediction or short fiber reinforced injection molded composites (5)
the measurement of the fiber orientation field develop- provided the fiber orientation is well characterized.
ing in injection molding. The measurement of fiber In modeling the flow of injection molded composites,
orientation is a cumbersome and labor intensive task; it is usual to consider the fiber orientation evolution
thus the need to explore the capabilities offered by as decoupled from the flow simulation (7). Conse-
new computer modeling tools follows naturally. quently, the fiber orientation field is not taken into ac-
The studies of Jeffery (2).who laid the basis for cur- count to determining the flow dynamics or to calculat-
rently available simulation packages, are valid only for ing the material viscosity, which can lead to important
very dilute suspensions. Dilute suspensions are those prediction errors. The improvement of the models by
in which the distance between neighboring particles using coupled solutions was attempted by Kabanemi
is, on average, bigger than their length. Most of the et aL (8)in order to obtain 3D fiber orientation predic-
commercially interesting composites do not fit to this tions. Their formulation includes the coupling be-
dilute regime. For other regimes, interactions between tween the fiber-orientation and the viscosity using an
neighboring fibers occur during flow and affect the earlier model proposed by Metmer (9).
final pattern of fiber orientation. I t is intuitive to ac- in commercial flow simulation packages, like C-Mold
cept that the neighboring fibers hinder the free rota- and Moldflow, it is possible to model the fiber orienta-
tion of a fiber. tion in each element of the model after the definition
Research work led by Tucker et al. (3-5).based on of the thermomechanical history. Most of these com-
experiments conducted with chopped fibers immersed mercial tools are based on the implementation of the
in silicone oil, led to a model that enabled accounting Folgar-Tucker model (3).This model includes a fiber

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Role of lnteraction Coeflicient

interaction coefficient C, in an added-in diffusion from second- to fourth-order requires the solution of a
term. This approach was successful in predicting with closure problem (5).A number of closure approxima-
reasonable accuracy the fiber orientation in plates tions have been proposed and applied with varying
and circular discs molded with heavily reinforced degree of success (5, 11). Moreover, the tensors offer
composites ( 10). the possibility of representing the statistical distribu-
In this work a study was conducted to analyze the tion of fiber orientation, and may be assigned to a
effect of the fiber interaction coefficient C, on the pre- physical meaning. The second-order tensor of fiber
dictions of the fiber orientation developing in lO?h by orientation has 3 X 3 terms (like a 3 x 3 matrix). The
weight glass fiber reinforced polycarbonate discs. The sum of the diagonal terms of the tensor is equal to the
orientation predictions were compared with experi- unity, and each of the diagonal values is allowed to
mental data obtained at three positions along the flow vary between 0 and 1. The value of each of the diago-
path of circular discs (radial direction). nal values stands for the relative orientation around
one of the co-ordinate axis. The axes are commonly
2. THEORY defined as 1-radial, 2-tangential, and 3- the out of
plane (or thickness) direction. The tensor being sym-
2.1. Definition of the Fiber Interaction Regime
metrical by definition implies that only three out-of-
The interaction regime in the composite melt is as- diagonal elements need to be computed. Therefore, five
sociated to the aspect ratio of the fibers (Lid), and to calculations are required to define the second-order
its volume fraction, (Vf). Typical long and slender tensor, without further simpliijang assumptions (only
fibers have an aspect ratio between 15 and 40. A di- two of the diagonal values are independent).
lute solution where the fibers hardly interact with The elements of the second-order tensor have a
each other is characterized by Vf < ( d / L ) 2(3).No com- physical meaning. The value of each of the diagonal
mercially important composites fall into this dilute elements of the tensor stands for the relative orienta-
regime. tion in one of the co-ordinate axes. The flow of poly-
If the volume fraction of fibers is inside the range mer composites in injection moldings is planar, lead-
( d / L ) 2< Vf < ( d / L ) ,the suspension is semi-concen- ing to planar fiber orientation states. Thus small
trated, and the spacing between fibers being smaller values of the a33tensor component are typical ob-
than L is larger than d The interaction between fibers tained. The element a l l is close to the unity if high
is frequent. The 10% by weight short-glass-fiberpoly- orientation in flow direction exists. In planar and ran-
carbonate composites used in this study fit into this dom distributions of fiber orientation the values of a ,
semi-concentrated regime. and aZ2are close to 0.5, a schematic representation
A highly concentrated solution is defined when V, > being shown in Fig. 1.
( d / L ) and the spacing between the fibers is of the In this case the second-order tensor is:
order of magnitude of the fiber diameter, d.
2.2. Tensors of Fiber Orientation
The state of fiber orientation is well described by the
ai,= [ ; ;]
0.5 0
0;
0

second- and the fourth-order tensors of fiber orienta-

tion (5, 10). The fourth-order tensors are required to
2.3. Modeling the Fiber Orientation
predict both rheological and mechanical properties of
fiber suspensions, because these are fourth-order ten- When the fiber concentration in the composite cor-
sor properties. responds to the concentrated or semi-concentrated
The use of the second-order tensor is convenient in regimes, it is important to determine the effect of the
terms of calculation time and is preferentially applied interactions between neighboring fibers of the fiber
in most of the commercial codes. The transformation orientation. The interaction coefficient, C,, depends on

I
/ \

', \ /

/ \
\
/ /
t"'
\ '
Fig. 1. Random planarfiber orientation.

POLYMER COMPOSITES, JUNE 2003, Vol. 24, No. 3 359

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A. J. Pontes, N. M. N e v e s , andA. S. Pouzada

the fiber aspect ratio, L/d,and on the volume frac- Table 1. Molding Program.
tion. Some authors suggest that the interaction coeffi- Code Melt temp. Injectionflow rate Mold temp.
cient should be dependent on the state of orientation ("C) (cm3I s ) ("C)
(12, 13). This statement apparently contradicts the
280.10 9.7
concept of the interaction coefficient being an intrinsic
property of the suspension, and brings an additional 280.14 280 13.8
transient character to the coefficient. However, with-
280.32 32.2
out a detailed model for describing the transient inter-
actions between fibers, there is no way to predict C,, 300.14 300 13.8 100
and therefore it must be determined experimentally. 320.10 9.7
Tucker and Advani (4) suggested a n empirical expres-
sion to be used when an experimentally determined 320.14 320 13.8
interaction coefficient is not available: 320.32 32.2

Two specimens per molding condition were randomly

This expression is valid for concentrated regimes selected to measure the fiber orientation distribution
(V,.L/d > 1). along and across the flow direction. For every molding
High values of the interaction coefficient lead to condition the fiber orientation distribution was experi-
random orientation states, whereas low values are mentally characterized at three points along the flow
likely to correspond to highly aligned distributions of path: a t 20 mm, 35 mm and 50 mm from the gate
fiber orientation in the flow direction (5). (Fig. 2).
The surfaces of samples cut from fiber reinforced
3. EXPERIMENTAL parts were subjected to successive stages of polishing
3.1. Molding (sandpaper of progressively smaller roughness) to
achieve a microscopically smooth surface. The fiber
The experimental study was done using a circular orientation was measured using image analysis tools
center sprue-gated disc, 1.5 mm thick and 114 mm in in images obtained by reflection microscopy of the
diameter. This geometry implies a radial divergent polished cross sections, using the method proposed
flow. A 10% by wei@t glass fiber reinforced grade of by Bay and Tucker (14). For the measurements, the
polycarbonate was used (Lexan 500 R from GE Plas- cross section was divided into three columns of twelve
tics). The manufacturer compounds this standard layers each with equal thickness to assess the sam-
grade material with a release agent and a flame-retar- pling error.
dant additive. The choice for this material was meant
to produce a moderate level of interaction between 3.3. From Measurements to the
neighboring fibers during flow, therefore replicating Fiber Orientation Tensor
the situation of a semi-concentrated regime.
The molding was done in a molding cell based on a The sections of the fibers appear in the images (Hg.
600 kN Krauss-Maffei 60/210 A injection molding 3 ) as circles if the fibers are aligned in a direction per-
machine. The molding program ( T a b l e 1 ) was devised pendicular to the cut section (0 = O'), rectangles if the
to study the effect of melt temperature and flow rate fiber axis is parallel to the section (0 = 90') or ellipses
on the fiber orientation of the moldings. otherwise.
The out of plane orientation angle, 0, is derived from
3.2. Fiber Orientation Measurement the major and minor axes of the ellipse, a and b, as

(3
The measurements of fiber orientation were made
0 = arc cos - (3)
on polished cross sections cut from the molded plates.

Fig. 2. Sections cutfrorn the discfor experimentalfher orientation characterization.

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Role of Interaction CoeBccient

Because the direction of penetration of the fiber in included in the FEM model. The mesh used in the
the specimens is not fully defined in a single cross C-Mold simulations is shown in Fig. 4.
section there is some uncertainty about the angle +, The information required to perform the simulation
+
indistinguishable from the angle + 180".This ambi- includes:
guity that affects only the out-of-diagonal tensor com-
The mesh of the part;
ponents is present in our data.
The in-plane orientation is determined by the angles The material properties:
defined by the major axis of the ellipse and the prese-
lected reference axes (Fig. 3 ) . These angles, 8 and +, The process conditions:
can be determined by digitizing the coordinates of the The parameter data.
endpoints of the major and minor axes of the ellipses,
The processing conditions were defined according to
either manually or by image analysis tools.
the set up for the experimental tests. The required
The relevant components of the unit vector p for the
material properties were obtained in the material
calculation of the orientation tensor are shown in Eqs
database of C-Mold, except for the number average
4 to 6 for the specimens cut in the 1-3 sectioning
fiber length, diameter and fiber volume fraction. These
plane.
data were determined experimentally after pirolysis of
p 1 = sin 8 cos + (4) polycarbonate granules, as 238 km, 14 pm and 5.8%,
respectively.
p2 = cos 0 (51 In order to increase the accuracy of the numerical
solution, the default number of layers across the
p 3 = sin 8 sin + (6) thickness was increased from the C-Mold standard
number of 12 to 20.
The elements of the second-order tensors that de- There are two main parameters to be adjusted when
scribe the fiber orientation can be calculated from the fiber orientation is simulated with C-Mold: the inter-
values of the vectors piusing Eq 7. action coefficient, C,,and the inlet boundary condi-
tion. In this study the boundary condition (at the
sprue entrance) was defined considering that the
fibers are aligned in the flow direction at the skin, and
randomly aligned at the core. This parameter is less
important when compared to C,,since at a short dis-
The parameter F, is a weighting function that corrects tance after the gate entrance the effect of this parame-
the bias resulting from the lower probability of a fiber ter on the simulation results vanishes. The inlet
lying parallel to the section plane appear in the image boundary condition is very important for large gates
(14). in which the orientation at the entrance dominates
the flow pattern.
3.4. C-Mold Simulations
3.5. Analysis of the Influence of C,
For the simulation of the fiber orientation a model
with 707 elements was created in the C-Mold pack- The effect of varying the interaction coefficient, C,
age (15).The direct sprue and the impression were was determined quantitatively considering the average

Fg.3. Possible forms of fher sections in a polishing surface, 4 is the out of plane orientatwn angle.

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A. J . Pontes, N. M. Neves, and A. S. Pouzada

Rg. 4. Mesh used to model thef h e r orientation.

deviation between the experimental and the predicted the all tensor component of fiber orientation higher
values for the a l l tensor component. For each set of than 0.6. At the core, the fiber alignment is high in
processing conditions the calculation uses data from the across flow direction both in the simulations pre-
three locations along the flow path (20,35and 50 mm dictions and in the experimental data, as it is sug-
distance from the gate) and twelve locations across gested by the large value of aZ2(small value of a , ,).
the thickness: The experimental results suggest a less defined lay-
ering through-thickness. At most of the data points

Deviation =
i2
I= I J- 1
lall,re.,,
(i.j)- (i.j)
alle,,,> 1 (8)
the a l l tensor component is lower than 0.5. These re-
sults c o n f m that the disc geometry, having a diver-
3 x 12 gent flow, induces higher fiber orientation in the
across flow direction.
The subscripts exp and pred refer to experimental and The simulations with C-Mold seem to predict quali-
predicted results, respectively. The indices i and j refer tatively well the fiber orientation close or far from the
to the location along the flow path and to the layers gate. At intermediate locations (in this case, 35 mm
across the thickness, respectively. from the gate) the degree of fiber orientation in the
flow direction is overestimated regardless the molding
4. RESULTS AND DISCUSSION conditions as it is summarized in the Fig. 6.

The results of the experimental fiber orientation 4.2. Effect of the Melt Temperature
characterization are presented in terms of the a l l sec-
ond order tensor component. This tensor component Increasing the melt temperature (Fig. 7) leads to
characterizes the fraction of fibers aligned in the flow lower levels of fiber orientation in the flow direction.
direction. The C-Mold software predicts a predomi- This is in accordance with the expectations, since the
nantly planar fiber orientation pattern with very small flow-induced stresses acting in the melt are propor-
values for the out-of-plane component of the tensor of tional to the viscosity. Lower melt temperature in-
fiber orientation (a33= 0.005). Thus, the transverse- creases and thus the alignment of the fibers. Experi-
flow-direction fiber orientation at any point in the re- mental data shows that molding with higher melt
sults may be seen as the difference between a,, and
the unity la,, = l-all). through-thickness fiber orientation (all -, -
temperatures results in close to random in-plane
0.5).
The relatively better agreement between predictions
and experimental data that are observed at the lower
4.1. Variation Along the Flow Path
melt temperature conditions (280.10, 280.14 and
Both experimental data and predictions correspond- 280.32) may be attributable to the stronger stress
ing to the processing condition code 280.10 considering field built up in the melt during the flow. Moreover, it
different C,are shown in Q. 5, at three locations along seems that in C-Mold the effect of shear stresses are
the flow path. The C-Mold predictions show a skin with more effectively predicted than the effect of exten-
fiber orientation predominant in the flow direction, with sional stresses.

362 POLYMER COMPOSITES,JUNE 2003, Vol. 24, No. 3

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Role of Interaction Coefficient

4
Flg. 5. Prediction and experimentalfber orientation in theflow direction at 280°C melt temperature and 9.7 cm3/sfzow rate, at three
distancesfrom the gate: a) 20 mm, b] 35 mm and c) 50 nun

4.3. Effect of the Flow Rate and 280.32). This is understandable because the
At the lowest melt temperature (Fig. 8) a clear ten- shear flow is better modeled and probably overesti-
dency of the fibers to get more aligned in the across mated by the C-Mold software. As a general observa-
flow direction is observed (all < 0.5). A slower flow tion it can be said that the effect of C , on the predic-
rate seems to favor the fiber orientation in the flow di- tions is qualitatively important, but it is not possible
rection. The fiber orientation in the flow direction is to find a value of C, that might lead to general qualita-
overpredicted close to the wall and at the core of the tive and quantitative close prediction of the fiber ori-
moldings. entation in the moldings.

4.4. Effect of the Interaction Coefficient 5. CONCLUSIONS

in the Simulations The interaction coefficient, C,,which is usually a
For the different processing conditions the effect of parameter to be defined by the user when modeling
C, in the predictions is shown in Fig. 9. the fiber orientation field in injection moldings, has an
As can be also seen therein, the deviation varies in- important effect on the qualitative prediction of the
versely to C,. The lowest deviation is obtained with C, fiber orientation. It affects significantly the patterns of
of 0.1. This is the value of C , that leads to an overall fiber orientation predicted in the flow simulations.
minimum deviation between experimental and pre- Larger values for C, tend to make the fiber orientation
dicted values of the tensor component all. Lower melt close to a random in-plane distribution whereas lower
temperatures appear to improve the prediction of the values predict a clear alignment in the flow direction.
fiber orientation pattern, as suggested by the smaller The fiber orientation field determined from experi-
deviations observed in these cases (280.10, 280.14 mental data in center gated discs molded using 10%

POLYMER COMPOSITES, JUNE 2003, Vol. 24, No. 3 363

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A. J. Pontes, N. M. Neues, andA. S . Pouzada

0.4 I

20 mm I n
- 0-

0 0
C ] = 0.1 0
0.3

. . . . . . . . . . . . . . .
0
0

0.2 0 -
B 0
0- ~~ . . . . . .

A A
A
n ,
0.1 I I l

280.10. 280.14 280.32 300.14 320.10. 320.14 320.32

Molding condition
FUJ. 6. Deviation between predictions and experimental results-for dgferent positions along theJow p a t k 20, 35 and 50 nun from
the gate.

C-Mold CI4.01 43C-MOM CI=O.W C-Mdd C1401 43-C-Mdd Cl=Q

................................................

-
0

..........................

-0.5 -0.25 0 035 0.5 -0.5 -0.2s 0 0.25 0.5

FrnctiorrPl distance to the mid+ FmcfJOIIPI distance to the mLlpbrne
(4 (b)

0.8 C-MoldCI4.01 - D C M . Y C I = O .

0.6
-
I

' 0.4 Fig. 7. Prediction and experimentalfiber orientation

in the$ow direction a t 13.8 cm3/smw rate and a t 20
mm from the gate, a t dLfferent melt temperatures:
0.2
a) 280% b) 300°C and c) 320°C.

0 ' I
-0.5 -0.25 0 0.25 0.5
Fnrtiomldwtance to the midplane

(c)

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Role of Interaction Coefficient

A Epnclcd.l -O-CM.UCI=O.l
0.8 -
CMbUCI=Wl -C)-C-MoUCI=O

0.6 - ............................................. ................................................

-
"
0.4 -
-
8

0.2 - 0.2 _ ....................... ........................

-0.5 -0.25 0 0.25 0.5 -0.5 -0.25 0 0.25 0.5

FnetiolrPl dbtancc to the FmctlorrPldetrureto the midplane

(a) (b)

0.8
C-MddCI=O.Ol -O-C-M&K.i=O.W

0.6
-
0.4 Fig. 8. Prediction and experimental f h e r orientation
in theflow direction at 280°C melt temperature and
0.2
at 20 mm from the gate, at differentflow rates:
a) 9.7 cm3/s, b) 13.8 cm3/s and c) 32.2 cm3/s.
0
-0.5 -0.25 0 0.25 0.5
Fractional distance to the midplane

(c)

l?g.9. Deviation between C-Mold predictions using duerent C, and experimental results.

POLYMER COMPOSITES,JUNE 2003,Vol. 24, No. 3 365

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A. J. Pontes. N . M.Neves. and A. S . Pouzada

glass fiber polycarbonate seems to be better predicted Department of Polymer Engineering of the Universi-
when a large value of C, is used. When the value of dade do Minho, Portugal. The authors also acknowl-
this interaction coefficient is decided from using the edge the assistance provided by the C-MOLD Europe
empirical expression suggested by Tucker (4). poor office at Enschede, The Netherlands.
predictions result for lightly reinforced materials in
this molding geometry. REFERENCES
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8. ACXNOWLEDGMENTS
The authors are indebted to IMAT-Institute of Mate-
rials, and the Center for Polymer Engineering of the

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