Rheology of short fiber filled thermoplastics

Rheology of Short Fiber Filled Thermoplastics TAMAL GHOSH,* MIROSLAV GRMELA,** and

PIERRE J . CARREAU

Centre d e Recherche AppZiquee sur Zes Polymkres (CRASP) DkpartGment d e Gknie Chimique

Ecole Polytechnique Montrkal, Quebec H3C 3A7 Canada

We examine the applicability of the conformation tensor to describe the fiber orientation and rheology of moderately concentrated fiber-filled thermoplastics subjected to large deformation flow. To retain computational simplicity, we assume a Newtonian matrix. We present a model that can account for orientation effects, Brownian motion, semiflexibility, and interactions through excluded volume effect, of the fibers. The model predicts a wide variety of rheological effects. We present predictions of steady shear viscosity, primary normal stress and the creep functions, as well as uniaxial elongational viscosity, due to the fibers. We have compared rheological data for 9.54 wt% carbon fibers in polyethylene and 30 wt% glass fibers in polypropylene, with the model predictions. By defining an “effective fiber concentration,” we have been able to correlate the model well with data. With fitting parameters from the steady state viscosity vs. shear rate data, we have been able to predict the steady state primary stress coefficient data as well as the creep data.

INTRODUCTION

he importance of fiber filled thermoplastics has T experienced a major growth in recent years. This is understandable because of the superior mechanical and thermal properties of the composites. A variety of isotropic and anisotropic fillers are available (1). The advantage of fiber filled materials arises from the fact that these are anisotropic, and hence their mechanical and physical properties are also anisotropic: the composite is stronger along the direction of the orientation of the fibers, and weaker along the transverse direction. Several excellent reviews have appeared on this subject (2-5).

To be able to process these filled thermoplastics optimally, we need at hand constitutive equations that can predict the effect of shear and elongation on the fiber orientation, as well as the mechanical and rheological characteristics of these systems. Once the fiber orientation is known, it is possible to predict the mechanical properties of the composite (6). Several analyses have been presented, which essentially predict the effect of shear/elongation on the fiber orientation and the extra stress tensor of such systems.

In this work, we present a simple description of the fiber orientation and rheology of fiber filled thermo-

A preliminary version of this work was presented at the 8th Annual Meeting of the Polymer Processing Society, New Delhi. India (March 1992). *Present Address: Engelhard Corporation, Technical Center, Gordan, GA 31031.

plastics using the conformation tensor to describe the fiber orientation. We follow the method introduced by Grmela (7). and its consequent develop ment (8), to develop a set of equations to describe the evolution of the conformation tensor and its relation- ship with the extra stress tensor. By directly amving at the evolution equation and the equation for the extra stress tensor on the conformation tensor level of description, we have avoided the use of closure approximations altogether, and thus, circumvented several numerical problems (9). We present predictions of steady and transient shear viscosity, normal stress and the fiber orientation, as well as uniaxial elongational viscosity. We compare steady, as well as transient shear flow data with model predictions.

PREVIOUS WORK

Perhaps the earliest work on this subject was presented by Jeffery (lo), who calculated the instanta- neous angular velocity of a neutrally buoyant ellip soid in an infinite Newtonian medium undergoing creep flow. Later, Bretherton (1 1) showed that the rotation of any body can be modeled as that of an equivalent ellipsoid. Batchelor (12, 13) developed a generalized equation for the hydrodynamic stress for a suspension of long axisymmetric rigid particles with negligible inertia, subjected to a constant strain rate.

Dinh and Armstrong (14) extended Batchelor’s (12) work to develop a rheological constitutive equation

144 POLYMER COMPOSITES, APRIL 1995, Vol. 16, No. 2

Rheology of Short Fiber Filled Thermoplastics

for semiconcentrated suspensions. They used Jeffery’s equation (1 0) with an infmite fiber aspect ratio, to calculate the fiber orientation. The fibers are aligned along the plane of shear in simple shear flow, and do not contribute to the steady state rheological properties. Bibbo e t a1 (15) incorporated a correction factor to account for the fiber thickness and obtained improved predictions.

Along a similar line of approach, an elegant analysis was presented by Folgar and Tucker (16). They proposed a phenomenological approach to model fiber rotations as a result of the competing effects due to interaction (diffusion) and imposed shear (convection). The convection term came from Jeffery’s equation (lo), and the interaction term was modeled as a term that randomizes the orientation distribution function as well as the flow field. An empirically introduced interaction coefficient incorporated the effects of the suspension parameters, uiz., fiber volume frac- tion, aspect ratio, etc. Kamal and Mute1 (17) followed a similar approach, but modeled the diffusion term as proportional to the viscous dissipation. An exten- sion of this and other prior work into a region of higher concentration has been discussed by Ran- ganathan and Advani (18).

Another approach using Jeffery’s model (10) and Batchelor’s analysis ( 12) has been presented recently by Ausias e t a1 (19). The model is developed for dilute suspensions of long rigid fibers and extended to concentrated suspensions. Only one parameter enters into the model. which characterizes the coupling be- tween stress and fiber orientation. Transient viscosity predictions are fairly good. They observed overshoots in transient viscosity predictions and attributed it to changes in fiber orientation.

A n entirely different approach was pursued by Be- craft and Metzner (20). Starting with a modified version of Doi’s molecular theory for liquid crystalline polymers (21), they adapted it to account for the macroscopic nature of the fibers. The distribution of fiber orientation was presented by means of a second order tensor. The predictions for steady shear viscosity and fiber orientations are very good.

We developed our model at the conformation tensor level. The conformation tensor provided a compact and physical representation of the fiber orientation (22, 23). We avoided the need for closure approximations and other numerical problems associated with other models (9).

DEVELOPMENT OF THE MODEL

The details of the development of the model are presented by Ghosh e t a1 (8); in this publication we concentrated on its application to fiber filled thermoplastics. The equations have been developed on the basis of the generalized Hamiltonian structure sug- gested by Grmela (7).

We consider the conformation tensor (a symmetric, positive-definite tensor) to represent the orientation of the fibers in three-dimensional space. The contribution due to the fibers towards the evolution of the

conformation tensor, _C( t) , is given as (7, 24): -

where D/D t is the material time derivative, _ K ~ is the velocity gradient tensor, A is the mobility Gpecified later in Eq. 51, and .Y is the Gibbs free energy (specified later in Eq 4). We assume a spatially homogeneous system, i.e., _C does not depend upon position. We note that the evzlution of the conformation tensor is composed of a convective part (dependent on the flow field) and a dissipative (or diffusive) part.

The equation relating the extra stress tensor due to the fibers, to the conformation tensor is (7, 24):

-

(2)

where n is the fiber number concentration (in rK3). We specify a spatial constraint in terms that the

fibers are “inextensible-in-average,” and incorporate it into the free energy by means of Lagrange multiplier, p :

(3)

where R, is the mean fiber length.

as the sum of the contributions to it: We specify the components of the Gibbs free energy

(4)

where k , is the Boltzmann constant, To is the constant absolute temperature and _S is the Kronecker delta. The four terms in Eq 4 resp&ively account for the spatial constraint, and the three components comprising the entropy functional (8): “Brownian motion,” “fiber semiflexibility,” and “excluded volume.” The quantities “S,,” “S,,” and “So” are parameters that respectively, weigh the three contributions to the entropy functional. The parameter S, weighs the dif- fusivity of the fibers. We note that the macroscopic fibers that we consider will not be significantly affected by diffusion. As we shall see later, the viscosity data for fiber filled thermoplastics is fitted by values of S, less than unity [for polymer melts and solutions, we hold S, equal to unity (811. The parameter S, weighs the Lifshitz contribution to the entropy functional and characterizes the “semiflexibility” of the fibers. The parameter So stems from Onsager’s analysis (25) of the “excluded volume” of macre molecules (or fibers in this case), and depends upon the concentration of the fibers, their average diameter and length.

We suggest the following expression for the mobil-

POLYMER COMPOSITES, APRIL 1995, Vol. 16, No. 2 145

Tarnal Ghosh, Miroslav Grmela, and Pierre J. Carreau

ity (8):

where A, is a constant, n, is the maximum-packing number concentration of the fibers (equal to 2n/3d2R,) and a is a parameter that we call the orientation parameter (0 I a I 1). The parameter a essentially serves as a correction to the effect of fiber orientation on mobility. As we shall see later, the parameter a controls the slope of the non-Newtonian asymptote of the steady shear viscosity vs. shear rate curves. Equation 5 accounts for the fact that the mobility of the fibers increases as either their concentration decreases or the fibers become more oriented along the direction of the axis.

The Lagrange multiplier, p , is obtained from Eq. 1 and 3:

r

1

At this point, we have introduced all the parame ters in the model that can be independently obtained. The parameter A, can be obtained from Eqs 7 and 1 1. The specification of the parameters a, S,, S,, and So will be discussed in the comparison of experiments. We now define a quantity, A, which will be shown later to possess the physical meaning of a time constant:

The time constant lumps together several characteristics of the fibers, viz., concentration, length, and mobility, and provides us a convenient means to compare the model predictions to the experimental data.

The contribution due to the polymeric matrix can be accounted for by say, considering another conformation tensor to characterize the orientation of the macromolecules in the matrix. To retain computational simplicity, we consider the polymer matrix to contribute to the extra stress tensor as:

where, g refers to the sum of the contributions to the extra stress tensor due to the fibers, g(f'bers), and the polymer matrix, _a(matrix). In the moderately concentrated solutions -that we chose to experiment, the steady shear viscosity and the steady shear primary stress coefficient of the polymer matrix were small with respect to that of the composite, except at the higher shear rates. Thus, as a first approximation, we assume the matrix to act as a Newtonian solvent, so that:

(9) (matrix) = - (matrix)- g12 ?lo Y

where gigatrix) is the 1 - 2 component of pcmatrix', is the zero-shear viscosity of the polymer

matrix, and y is the shear rate. One of the advantages of the Hamiltonian formula-

tion is the flexibility that enables us to develop models suited to our needs (26). By reinterpreting the various parameters, or choosing appropriate terms for the mobility and free energy, it is possible to describe the rheology of several complex polymeric systems. Starting with dilute polymer solutions (8), it has been possible to describe the rheology of melts, electrolytes (23). and then fiber filled thermoplastics.

As to the limits of applicability of the model, it should best be ascribed by actual experiments.

We note that an analog of the model by Becraft and Metzner (20) can be recovered from our model by an appropriate choice of the parameters: a = 0, S, = 1 , S, = 0, and So = ndR5 (or v 1 = l), where d is the fiber diameter and v 1 is a numerical constant.

MODEL PREDICTIONS

Solution Strategy

We wish to restrict ourselves to predictions of simple shear and uniaxial elongational flow. We have a pure rheological problem at hand, i.e., the flow field (27) and its magnitude is specified. We look for solutions of the five simultaneous autonomous ordinary differential equations describing the evolution of the conformation tensor ( E q 1 ) and then simultaneously solve the equation relating the internal structure to the extra stress tensor ( E q 2).

The highly nonlinear and complex nature of the evolution Eq 1 dissuaded us from attempting analytical solutions. For the choice of parameters as: a= 0, S, = 1, S,= 0, and So= 0, we have analytical solutions in both simple shear and elongational flow (281, and used these to check the numerical algorithm and control the errors.

The initial condition for the conformation tensor (corresponding to t = O and no-flow) is given by 6 5 / 6 _ C = 0. This is again a highly nonlinear tensorial equation and can exhibit multiple solutions, which indicate the possibility of multiple phases existing in equilibrium (26). I t is also possible that the initial condition is not in equilibrium: we follow the a p proach of Ausias et aL (19) and account for it by

146 POLYMER COMPOSITES, APRiL 1995, Voi. 76, No. 2


considering a diagonal tensor: I ' ' """I ' ' """I ' ' """I ' ' """I ' ' 'l 'rlr(

where Cinit is a parameter. In the absence of any preferred orientation (isotropic condition), Cinit = 3. As such, Cinit may vary from 0 (total orientation along the plane normal to the shear plane) to 1 (total orientation along the shear plane).

We choose a fifth order Runge-Kutta method (29) to solve the evolution equation numerically. The material functions are defined according to Bird et aL (27). We note that this model is unable to predict the secondary normal stress coefficients (8).

Perturbation Analysis

We obtained the limiting values for the material functions by expanding the extra stress tensor and the conformation tensor as polynomial expansions in shear/elongation rate, substituting them in the E q 1 and 2, and make the shear/elongation rate go to zero:

1. fiber contribution to the zero-shear viscosity,

(11)

2. fiber contribution to the zero-shear primary nor-

77fbers): 77&ffbers) = Ank B O T

mal stress coefficient, +$?):

3. fiber contribution to the zero-elongation viscosity,

(13)

a b e r s ) : (fibers) = 3 ~ ~ k T - 3 (fibers) % B 0- 770

Predictions of the stress relaxation function (27) at low shear rates from the model show that it is proportional to exp( - A t ) (281, and hence the quantity A has the physical meaning of a time constant.

Prediction of Fiber Orientation and Material Functions

We present plots of steady shear viscosity arising due to the fibers in Qs. 1-4. We notice that the effect of the orientation parameter, a, is to vary the non-Newtonian slope of the viscosity curves. For a choice of parameters (typical for fiber filled systems) as S, = 0.6, S, = 0.0, and So = 1.0, the slope corresponding to a= 0 is -0.66, whereas a= 1 corresponds to a slope of -0.86.

For the limiting case of SB = 0, we note that the slope is approximately -0.50 (Fig. 2). For both S , = 0 and S,= 0, the evolution equations fail to converge. In Fg. 3, we note that the effect of the semiflexibility

100

10' 0

F F

1 0 2

1 0 2 1 0' 1 00 10' 102 103

FYg. 1. Steady shear viscosity as a function of shear rate, due to _fibers: varying a (orientation parameter). S, = 0.6, S,=O.O,S,= 1.0, a n d C i n i t = $ .

I ' ' """I ' ' """I ' ' """I ' '"""I ' ' """I ' '"'I 100

10-1

10-3

-

SpO.0,

: S,=l.O, S,=O.B,

I S,-0.6, -

100 4

1 0 2 lo-' 100 10' 1 02 103

k;( Fig. 3. Steady shear viscosity as afunction of shear rate, due toftbers: varying S, (Lifshitz parameter). a = 1.0, S, = 0.6, So = 1.0, and Cinit = 5.


Tarnal Ghosh, Miroslav Grmela, and Pierre J . Carreau

I ' ' " l ' . ' l ' ' """ ' ' ' " " " ' ' ' """1 1 00

10' 0

F F

10"

10-1 1 00 1 0' 1 0' 103

x i Fig. 4. Steady shear viscosity as a function of shear rate, due tofibers; varying So (Onsagerpararneter). a= 1.0. S , = 0.6, S, = 0.0, and Cinit = 5.

10-2 lo-' 1 00 10' 1 02 103

li;u Fig. 5. Primary normal stress coefficient as a function of shear rate, due to Jbers; varying So (Onsager parameter). a = 1.0, S, = 0.6, S, = 0.0, and Cinit = 5.

parameter is to delay the onset of non-Newtonian behavior and decrease the shear-thinning behavior. We recall that Blakeney (30) has reported that a slight curvature in the fibers results in a significant increase of the relative Newtonian viscosity of a suspension of fibers. In Fg. 4 we note an interesting effect of the excluded volume or the "Onsager parameter," So. With increasing So (proportional to the fiber concentration), we see occurrence of yield stresses. This is in qualitative agreement with experimental observations, though we are yet to analyze it quantitatively. However, we also observe shear- thinning behavior for a small value of So (proportional to concentration). This is related to the pres- ence of the Brownian motion term, that we are forced to overemphasize in our model. We note from this figure that at the higher shear rates, the viscosity curves tend very close to each other, an observation consistent with experimental observations (2, 3 1).

In Fig. 5 we observe predictions of the fiber contribution to the primary normal stress coefficient, I/I1. We can estimate the zero shear stress coefficient, 1/11,,

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

tm. Fig. 6. Creep function due to Jbers, with varying "So" (Onsager parameter). a = 1 .O, S, = 0.6, S , = 0.0, and c . . = I

Lrlll 3 .

1 .o 0.9

0.8 0.7

iZ$ ;; 0.3

0.2

0.1

c,,=9/10 ci"i,=lll 0

cin,=112

. . . - . . . . Ci,,=1/3

- - /

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 f f h

Fig. 7. Creep function due to Jbers, with varying "Cini," (initial condition). %= 1.0, a= 1.0, S,=0.6, S,=O.O, and so = 1.0.

from the model, though it is quite difficult to measure it experimentally.

We present the transient material function, the creep function due to the fibers, for a given imposed shear stress, go, next. In Fig. 6 we clearly observe the onset of the inverted creep behavior (corresponding to overshoots in transient viscosity predictions) due to orientation effects, as documented by Ausias et aL (19). The inverted creep behavior sets in with increasing stress level at the same fiber concentration (proportional to So), and is a very good qualitative description of actual experimental observations. In Fig. 7, we observe the effect of the initial condition by means of the parameter Cinit. For values of Cinit close to the isotropic condition (equal to +), the behavior at small times is only slightly affected, and at steady state, the predicted creep values approach the same slope (or shear rate). However for Cinit= s, where the fibers are almost perfectly oriented along the plane of shear, we notice a region of high shear rate, followed by a region of low rate of shear (corresponding to overshoots in transient viscosity), and



U 1 00

10-1

10-2

1 n-3

S,=O.Ol S,=lO.O

I"

10' 100 10' 102 103

L? Fig. 8. Steady state Jber orientation predictions, effect of shear rate and "So" (Onsager parameter). a = 1 .O, S, = 0.6, and S, = 0.0.

ultimately a steady state. Since the fibers are highly oriented along the plane of shear initially, they offer less resistance to flow. I t is possible that the maximum in the initial region of the shear corresponds to a change in structure. We also notice that the steady state slope achieved by this curve is much different from the other curves (with initial condition close to the isotropic condition), which means a different steady shear viscosity, indicating to us the possibility of a separate phase (26) or "clusters" (32).

Next we present the predictions of fiber orientation described by the conformation tensor. We map the conformation tensor on a two-dimensional plane, the horizontal axis representing the plane of shear. The major axis of the ellipse represents the preferred direction, and the minor axis is an indication of the spread in the distribution. The steady state predictions are presented in Fig. 8. At low shear rates and for very little excluded volume of fibers (So = 0.0 l), we observe that there is a considerable spread in the preferred direction, but the spread decreases with increasing shear rates. However for a higher excluded volume (So = 10). we observe a sharp preferred orientation even at low shear rates, and the orientation does not change much with increasing shear rates. This observation is consistent with the analysis of Bibbo and Armstrong (33), as well as Ranganathan and Advani (181, but contradicts the results of Kamal and Mutel (7, 31). The transient fiber orientations are presented in Fig. 9. At time, t= 0, we start with an isotropic distribution (Cinit = 4). and upon the impo- sition of shear, the fibers quickly orient themselves along a preferred direction. The preferred direction does not reverse with increasing time. This is consistent with the observations of Kamal and Mutel (17). However, we note that our model predicts that on the cessation of shear, the fibers regain their original isotropic orientation distribution. This is because the role of Brownian motion of the fibers is overempha- sized in our model.

The effect of the parameter So on the steady state

uniaxial elongational viscosity of the fibers is Fig. 10. The elongational viscosity generally decreases with increasing elongational rate, except for a zone of constant elongational viscosity and a zone of a slight increase for So = 0.01. This is consistent with experimental observations as well as physical reasonings (1 7).

COMPARISON WITH EXPERIMENTS

We performed simple shear flow experiments with short glass fiber filled polypropylene. The material was supplied as cylindrical pellets of 30 wt% short glass fibers in polypropylene (Appryl 34001, by Elf Atochem. We prepared the samples by compression molding (at -20 MPa and 180°C for 10 min). We used a Rheometrics RSR Constant Stress rheometer and a Bohlin CSM rheometer, both in parallel plate configuration; the gap was held constant at 1.5 mm and the diameter of the plates were 25 mm. Both the rheometers were operated in the creep mode. For the Rheometrics rheometer, we could vary the imposed

" A 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5 . 0 5 6 7

0 tll d Fig. 9. nansient f&er orientation predictions. h j = 3.33, a =

1.0,SB=0.6,SL=0.0, andSo=6.0.

I . ,...,,. 1 , . I ..... I , ,.,..,., I ......,, , ..,,..,, , . . , ,

104 10-3 10-2 10-1 100 101 102 103

G Fg. 10. Steady uniaxial elongational viscosity as a function of elongation rate; varying So (Onsager parameter). a = 1 .O, S, = 0.6, S, = 0.0, and Cinit = 5.


Tamal Ghosh, Miroslav Gmela, and Pierre J . Carreau

shear stress from 50 to 500 Pa, and for the Bohlin rheometer from 1 to 2000 Pa.

The major experimental difficulties we faced were in trying to obtain a homogeneous fiber distribution in different samples, and possible thermal degradation of the samples during measurement. We chose only those samples that seemed to show an uniform fiber distribution by visual observation. To impede the rate of degradation we used nitrogen gas as the heating fluid in the convection chamber of the Rheo- metrics rheometer. In the Bohlin rheometer, heating is done electrically and is more precisely controlled. In this rheometer too, we passed a slow stream of nitrogen gas through the convection chamber.

We considered another set of data for carbon fibers in polyethylene from Sakai et al (34); steady shear viscosity and first normal stress data for 9.54 wt% carbon fibers in polyethylene.

Once we estimate the zero-shear viscosity of the composite and the polymer matrix from simple shear flow experiments, and specify the parameter a (from the asymptotic slope of the steady shear viscosity vs. shear rate data), we have fixed all the parameters in the model, except those contributing to the entropy ( S B , S,, and So>. The physical characteristics of the two composites, obtained independently of the rheological experiments, are listed in Table 1. With these parameters, and fitting S,, S,, and So, we can predict the steady state viscosity vs. shear rate function. However, this exercise proved futile: we noticed a

good qualitative prediction, but the shear rates were under-predicted by several orders of magnitude (1 O9 for 30 wt% short glass fibers in polypropylene and 10’l for 9.54 wt% carbon fibers in polyethylene).

To deal with this, we chose to consider the number concentration of the fibers, n, as an “effective fiber number concentration,” n:ff: a parameter whose magnitude is obtained by fittmg experimental data; steady shear viscosity vs. shear rate. Becraft and Metzner (20) have used a similar idea to define an effective flber concentration. This reinterpretation accounts for the fact that our model was originally introduced to deal with macromolecules, but in this case we are dealing with macroscopic fibers. The quantity neff is several orders of magnitude higher than the number concentration, n, of the fibers. We decided to stay with the quantity neff because it exhibits a clear physical meaning, viz., the effective number of fibers per m3 of the composite, inspite of the fact that its order of magnitude is uncomfortably large (see Table 2).

We now introduce an “effective time constant,” A, ff, and redefine Eq 1 1 as:

The fitted and the calculated model parameters for the two composites are presented in Table 2.

mure 1 1 compares the predictions and the data

Table 1. Physical Characteristics of the Two Composites.

9.54 wt% Carbon Fibers in Polyethylene

@200”C (34)

30 Wrro Short Glass Fibers in Polypropylene

@200”C

Mean fiber length, R , 2.66 x m 2.55 x m Fiber diameter, d 7.0 x m 1.0 x m Fiber aspect ratio, R /d 38 26 Density of fibers, ptftgers) Density of matrix, p(matrix)

number concentration, n,

1.78 x lo3 kg/m3 7.56 x 10’ kg/m3

2.4 x lo3 kg/m3 7.39 x 10‘ kg/m3

Fiber number concentration, n 4.19 x 1012/m3 4.99 x 10131m3 Maximum-packing fiber 4.82 x 1014/m3 2.46 x 10141m3

Table 2. Fitted and Calculated Model Parameters for the Two Composites.

9.54 wt% Carbon Fibers in Polyethylene

1.02 x lo6 Pa.s

2.19 x lo3 Pa.s

30 wt% Short Glass Fibers in Polypropylene

2.43 x lo6 Pa.s

4.12 x 10’ Pa.s

@200°C (34) @200”C

Zero-shear viscosity of composite, 7, (fitted) Zero-shear viscosity of polymer matrix, qimatrix) (fitted) Orientation parameter, a 1 .o 1 .o Boltzmann parameter, S, 0.8 0.4 Lifshitz parameter, S, 0.0 0.0 Onsager parameter, So 1.5 0.9

Effective fiber number 5.21 x 10231~3 1.25 x 1021m3

Effective time constant

Mobility constant, A, 0.086 s/kg 0.413 s/k

concentration, ne ,, Time constant (equation 1 l ) , A 3.73 x 10’3 s 7.46 X 10“ s

3.01 x 10‘ s 2.97 x 105 s CEq 14), Aeff



for steady shear viscosity vs. shear rate for 9.54 wt% carbon fibers in polyethylene at 200°C (34). We fitted the data using the following parameters: a= 1.0, S, = 0.8, So = 1.50, and neff= 5.21 X 1023/m3. We are unable to predict a better slope of the non-Newtc- nian asymptote because it is not possible to exceed a= 1.0 and still retain a physical meaning. We wish to highlight the physical significance of each fitted parameter. The “high” orientation of the 9.54 wt% fibers is reflected by the high value of the parameter a Since Brownian motion becomes less important, we have S , less than unity. Carbon fibers are nearly rigid (34) and so we have taken S, = 0. The parameter So is proportional to the fiber concentration and dimensions.

Figure 12 compares the model predictions of the primary normal stress coefficient for the parameters used in Fg. 11. We do not have a value of ql, from the experiments, but predict it by using a modified form of Eq 12:

(15)

100

‘-4 10-2 10-1 100 10’ 1 02

1.L Fg. 1 1 . Steady shear viscosity as a function of shear rate: 9.54 wp/, carbonJbers in polyethylene at 200°C (34).

model fit (parameters from q vs. k f data)

10-2

10-1 1 00 1 0’ 102 103 104

k?

Q. 12. Steady shear primary normal stress coemient as a function of shear rate; 9.54 rut% carbonfibers in polyethylene at 200°C (34).

The calculated value was 3.40 x lo8 Pa.s2. The model predictions are high by about half an order of magnitude in the lower shear rate range, but get closer at the higher shear rates.

Next, we will fit the model parameters from the steady shear viscosity vs. shear rate data and predict the creep function. Figure 13 presents the steady shear viscosity data as a function of shear rate for experiments with 30 wt% short glass fiber filled polypropylene at 200°C. We could fit the data for: a= 1.0, S, = 0.4, So= 0.9, and neff= 1.25 X lo2’ m-3. The limit a= 1 again prevents us from obtaining a better prediction of the non-Newtonian asymptotic slope. The magnitude of the parameters here again reflect well their physical significance.

We use the parameters obtained from Fig. 13 to predict the creep functions at imposed shear stress of 10, 250, and 300 Pa, respectively. At a low imposed shear stress of 10 Pa, the model predicts the creep function quite well (Fig. 14A). The slight quantitative mismatch is not surprising at such low levels of shear. For a n imposed shear stress of 250 Pa (Fig. 14B). the model predictions agree very well with the experiments. At a slightly higher imposed shear stress of 300 Pa (Fig. 14C), we observe the inverted creep behavior [corresponding to overshoots in transient viscosity (1911 and our model is able to predict it reasonably well.

CONCLUSIONS

We have presented a mathematical model that predicts well the rheology and the fiber orientation of short fiber filled thermoplastics. The model has its origins in equilibrium thermodynamics. By choosing the conformation tensor to describe the fiber orientation in space and using the Hamiltonian structure to derive the evolution of the conformation tensor and the expression for the extra stress tensor, we have avoided the use of closure approximations and circumvented several mathematical problems. By fitting model parameters from the steady viscosity vs. shear rate experiments, we have been able to satisfactorily

1 00

lo” 0

F E:

102 Experimental data (Rheometrics Experimental data (Bohlin CSM)

Model fit for: SB=0.4, So-0.9, a=1 .O. n -1 .252x1OZ1mJ

n

1 q0=2.4x1 Os Pa.s

10-2 10 ‘ 1 00 10’ 10‘ 103

xu F‘ig. 1 3 . Steady shear viscosity as a fiction of shear rate: 30 WUO short glassJbers in polypropylene at 200°C.


Tarnal Ghosh, Miroslav Gm wla, and Pierre J. Carreau

1 .o 0.9 0.8 0.7 0.6 0.5 0.4

0.3 0.2 0.1

0.0

Experimental data (10 Pa)

Model (parameters fitted from q vs. Xy data)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 vh

(a)

1.0 , . ' , I . , ! , , . , . , , . . I . I . . , , I , , . I

0.9 Experimental data (250 Pa)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 tA

(b)

1 .o r.. , . . , ' " ' " ' F F ' ' ' " ' I " ' " ' " ' I ' ' " " ' " I ' " ' " ' ' ' I Experimental data (300 Pa) - Model (parameters fitted from q vs. X+ data)

0.2 0 . 3 ~ 0.0 0.1 ***

0.00 0.05 0.10 0.15 0.20 0.25 0.30

vh

(C )

FYg. 14. Creep function: 30 WUO short glass fibers in polypropylene at 200°C; a) imposed shear stress, uo = 10 Pa; b) a, = 250 Pa: c) uo = 300 Pa.

predict the primary normal stress coefficient as well as the creep function.

To retain computational simplicity, we have had to assume a Newtonian matrix. While this seems to be a reasonable assumption in the range of shear rates and fiber concentrations studied, it would be useful to incorporate the viscoelastic contribution of the

polymer matrix, and delineate the Brownian motion effects due to the fibers and the matrix.

In addition, we would like to remove some of the limitations of the model. An alternative expression for the mobility so as to enable us to traverse a greater range of the slopes of the non-Newtonian asymptote, would be very useful. Also, we would like to obtain an analytical expression relating the "effective" time constant to the properties of the polymer matrix and fibers.

ACKNOWLEDGMENTS

We are grateful to Prof. T. Kitano of the Research Institute of Textiles and Polymers, Tsukuba, Japan, for providing us with experimental data (34). We gratefully acknowledge the financial support received from the Natural Sciences and Engineering Research Council of Canada and from the Fonds pour la forma- tion de chercheurs et l'aide 21 la recherche (FCAR) of the Province of Qukbec.

NOMENCLATURE

a Cinit C d =

R O

sI3

t tr TO

= Orientation parameter. = Parameter (initial condition). = Conformation tensor. = Derivative. = Fiber diameter. = Exponential function. = Gibbs free energy. = Boltzmann constant (1.3805 X = Number concentration of fibers. = Effective number concentration of fibers. = Maximum-packing number concentration of

= Mean fiber length. = Parameter weighing Brownian motion

= Parameter weighing contribution due to fiber

= Parameter weighing contribution of excluded

= Time. = Trace of a tensor. = Constant absolute temperature.

J/K).

fibers.

contribution to entropy.

semiflexibility to entropy.

volume of fibers to entropy.

Greek Symbols

= Lagrange multiplier. = Shear. = Shear rate. = Kronecker delta. = Steady shear viscosity. = Steady elongational viscosity. = Shear viscosity growth function. = Shear viscosity decay function. = Zero-shear viscosity. = Zero-elongation viscosity. = Transpose of velocity gradient tensor. =Time constant. = "Effective" time constant.


Rheology of Short Fiber E

A =Mobility. A, = Constant (mobility). u1 = Constant. go

- U = Imposed shear stress in creep mode. = Extra stress tensor. = Primary normal stress coefficient. = Zero-shear primary normal stress coefficient.

-

Superscripts

fiber contribution. matrix contribution.

= transpose of a tensor. = inverse of a tensor.

(fibers) =

(matrix) =

T

- 1

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Received March 15, 1993 Revision received April 1994

Addison-Wesley, Reading, Mass. ( 1982).

(1985).

( 199 1).


Rheology of short fiber filled thermoplastics

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