Rheology and processing of suspensions with fiber ... - Springer Link

Korea-Australia Rheology Journal Vol. 23, No. 4, December 2011 pp. 219-226 DOI: 10.1007/s13367-011-0027-1

Rheology and processing of suspensions with fiber, disk and magnetic particles Jang Min Park1 and Seong Jin Park1,2,* Department of Mechanical Engineering, Pohang University of Science and Technology, San 31, Hyojadong, Namgu, Pohang, Gyeongbuk, 790-784, Republic of Korea 2 Division of Advanced Nuclear Engineering, Pohang University of Science and Technology, San 31, Hyojadong, Namgu, Pohang, Gyeongbuk, 790-784, Republic of Korea (Received August 9, 2011) 1

Abstract This paper introduces recent development and progress in rheology and processing of suspension materials. In particular, three topics with different suspensions will be discussed, which are relevant in polymer- and powder-based manufacturing industries for various applications. The paper begins with a fundamental study of rheological modeling for short fiber suspension in a viscoelastic liquid. Irreversible thermodynamics based approach is presented to describe the polymer viscoelastic deformation with the fiber anisotropy effect taken into account. Several distinguishing features of the model are presented. Then we turn our attention to more industry-oriented subject of pigment disk orientation in injection molding of aesthetic plastic parts. The major difference between fiber and disk orientation kinematics is briefly discussed using Jeffery model. Experimental and numerical results for the disk orientation and the surface color of the molded part are presented. Finally, we introduce our recent efforts regarding the rheology of magnetic particle suspension, which is an essential part for a development of precision magnetic powder injection molding process. Some rheometer data for stainless steel powder suspensions are presented. Keywords: suspension, particle orientation, powder injection molding

1. Introduction One can find various suspension materials in polymer- and powder-based manufacturing processes. A reinforced thermoplastic composite is one representative example of such material, where micro reinforcement particles are compounded with a neat polymer to enhance the thermo-mechanical property of the final product (Chan et al., 1978; Folgar and Tucker, 1984). Also there have been efforts to disperse nano-scale materials in a polymeric matrix for reinforcement or for modification of the electrical property of the polymer (Kagarise et al., 2011). In the powder-based manufacturing process, one can find the suspension materials where metal or ceramic powders are mixed with a polymeric matrix to ease the shaping of powder into a complex three-dimensional geometry. A powder injection molding (PIM) is one representative processing which uses the suspension material for a mass production of metal or ceramic products. In this case, the suspensions are commonly in a highly concentrated regime, thus they have special features such as yield stress and wall slip behavior (Kwon and Ahn, 1995). The processing with the suspension material requires *Corresponding author: [email protected] © 2011 The Korean Society of Rheology and Springer

Korea-Australia Rheology Journal

comprehensive understanding of the suspension rheology. This is quite important particularly for the suspensions with non-spherical particles, since the physical property of the final product will be significantly affected by anisotropic orientation and distribution of particles developing during the process. In this respect, various theoretical, numerical and experimental studies have been carried out in the literature. The short fiber suspension would be one representative example of which rheology and processing have been studied comprehensively in the past three decades (Chan et al. 1978; Advani and Tucker, 1987; Gupta and Wang, 1993; Phelps and Tucker, 2009). In this paper, we introduce three subjects with different suspension materials which are relevant in polymer- and powder-based manufacturing industries. In the first subject, a theoretical study is introduced for the rheological modeling of the fiber suspension particularly in a viscoelastic media (Park and Kwon, 2011a). The main feature of this model is a phenomenological coupling between fiber and polymer kinematics. The second subject is on a disk (or flake) orientation and surface color in injection molding of pigmented polymer composite (Park et al., 2011). This subject discusses on a recent progress for prediction of surface color property in injection molding of aesthetic metal- or china-like plastic parts. The third subject deals with rheology of magnetic par-

December 2011 Vol. 23, No. 4

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Jang Min Park and Seong Jin Park

ticle suspensions, which is a fundamental study for a development of precision- and micro-magnetic PIM process. This subject is a part of a project in progress, thus a brief overview of the project and some rheological data will be discussed.

2. Fiber Suspension in a Viscoelastic Media As far as the rheological modeling for the short fiber suspension is concerned, there are two major issues which are (i) the kinematics of the fibers and (ii) the bulk stress contribution due to the fibers (Batchelor, 1970). Based on a pioneering work of Jeffery on the kinematics of ellipsoidal particle in a viscous liquid (Jeffery, 1922), Tucker's group has investigated on the fiber kinematics in detail for past three decades (Folgar and Tucker, 1984; Advani and Tucker, 1987; Cintra and Tucker, 1995; Wang et al., 2008; Phelps and Tucker, 2009), and their development has been widely accepted for the fiber orientation prediction in injection molding of short fiber composite material (Gupta and Wang, 1993; Chung and Kwon, 1995; Chung and Kwon, 2002; Park and Kwon, 2011b). As for the bulk stress in the suspension, Batchelor (1970) derived a general expression of the stress contribution due to the arbitrary particles immersed in a viscous matrix using ensemble average concept. For non-dilute fiber suspensions, which are of practical interest in the engineering field, several constitutive model have been developed in the literature (Dinh and Armstrong, 1984; Shaqfeh and Fredrickson, 1990; Phan-Thien and Graham, 1991). In this study, we briefly introduce a rheological model for a short fiber suspension in a semi-concentrated regime particularly in a viscoelastic media, which has been recently proposed by Park and Kwon (2011a). Only major features of the model will be presented here and one can find more details from the original paper. The model is derived via irreversible thermodynamic description of the viscoelastic deformation, where fiber anisotropic effect is taken into account in a manner of positive entropy production. The final form of the stress tensor combines viscoelastic part of Leonov model and fiber anisotropic part of Dinh-Armstrong model. The kinematic equation of the reversible Finger strain tensor (ce) for a viscoelastic deformation could be written as follows (Leonov, 1976): ∇

ce = – e p ⋅ c e – c e ⋅ e p

(1)

where the left hand side is the upper convective time derivative of ce , and ep is the dissipative rate-of-deformation tensor. According to irreversible thermodynamics approach by Leonov (1976), Park and Kwon (2011a) proposed following form of ep : –1 1 1 ep = α1 C1⎛ ce – --- I1 δ⎞ – C2 ⎛ ce – --- I2δ⎞ ⎝ ⎝ 3 ⎠ 3 ⎠

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2 1 1 + α2 B1 ⎛ a – --- J1δ⎞ – B2⎛ ae – --- J2δ⎞ ⎝ 3 ⎠ ⎝ 3 ⎠

(2)

where a is the second-order fiber orientation tensor, δ is the identity tensor, I1 and I2 are the first and the second invariants of ce, respectively, and J1 ( = 1) and J2 are the first and the second invariants of a, respectively. It should be noted that the fiber anisotropic terms appear in Eq. (2) in addition to the original Leonov model terms. The α1 and α2 represent the contribution from polymer and fiber, respectively, to the dissipative rate-of-strain tensor. C1 and C2 are functions of invariants of ce , and B1 and B2 are functions of joint invariants of ce and a. For a simplicity of the model, C1 = C2 = 1 ⁄ (4θ ) was chosen where θ is a relaxation time of the polymer. Also it was assumed that B1 = B2 = B . From the second law of the thermodynamics, namely Clausius-Duhem inequality, Park and Kwon (2011a) proposed the following form of B: 2 1 1 B = ------ ( a : ce – a :ce ) – --- ( J1 – J2 )I1 3 4θ

(3)

For a kinematics of single ellipsoidal particle in a viscous liquid, the Jeffery model reads as follows (Jeffery, 1922): · 1 1 p = – --- ω ⋅ p + --- λ ( γ· ⋅ p – γ· : p ⊗ p ⊗ p ) 2 2

(4)

where p is the orientation vector of the ellipsoidal particle, ω is the vorticity tensor, γ· is the rate-of-deformation tensor and λ is the particle geometrical parameter which varies from –1 for disk-like geometry to 1 for a fiber-like geometry. One can obtain the evolution equation of a from Eq. (4) as follows (Advani and Tucker, 1987): Da λ ------- = – 1 --- ( ω ⋅ a – a ⋅ ω ) + --- ( γ· ⋅ a + a ⋅ γ· – 2A :γ· ) Dt 2 2 + 2γ· CI ( δ – 3a )

(5)

where the last term in the right-hand-side was introduced by Folgar and Tucker (1984) to take the fiber interaction effect into account for concentrated suspensions. CI is the fiber interaction coefficient, γ· is the generalized rate-ofstrain, and A is the forth-order orientation tensor. Finally the stress tensor combining the viscoelasticity of polymer and fiber anisotropy is expressed as follows: ηp σ = – pI + ηs γ· + ηm Np γ· :A + -----c e θ

(6)

where p is the pressure, ηs is the solvent (for polymer) viscosity, ηm is the viscosity of the viscoelastic media and ηp is the polymer viscosity. Np is a particle number which represents the contribution of fiber anisotropy, and Dinh-Armstrong model reads as follows (Dinh and Armstrong, 1984): Korea-Australia Rheology Journal

Rheology and processing of suspensions with fiber, disk and magnetic particles

Fig. 1. The effect of fiber fraction on (a) the shear viscosity and (b) the first normal stress difference. Fig. 2. Fitted model curves and experimental data for neat and fiber reinforced polystyrene (a) the shear viscosity and (b) the first normal stress difference.

3

πnf L Np = --------------------------12 ln ( 2h ⁄ D )

(7)

where nf is the fiber number density, h is the average space between neighboring fibers, and L and D are fiber length and diameter, respectively. Comprehensive parameter study has been carried out with the proposed model, and major features of the model will be discussed here. Fig. 1 shows the effect of fiber volume fraction ( φ f ) on the shear viscosity ( η ) and the first normal stress difference ( N1 ). The shear viscosity increases with φ f , and it is more significant particularly at the lower shear rate regime. Also the shear thinning tends to start appearing at a lower shear rate as φ f increases. In the high shear rate regime, the effect of φ f on η becomes relatively small, and the viscosity curves are almost parallel with each other for various values of φ f . The first normal stress difference also increases with φ f in the overall range of the shear rate. Also the curves for different values of φ f are parallel with each other in the high shear rate regime. Fig. 2 is the comparison between the Korea-Australia Rheology Journal

model prediction and the experimental data of fiber reinforced polystyrene melt by Chan et al. (1978). Model parameters were determined by least square fitting to the experimental shear viscosity data. Two relaxation modes were used to represent the highly non-linear nature of the thermoplastic melt. The model can predict the experimental result of the shear viscosity accurately in the overall range of the shear rate for three different fiber volume fraction cases. The shear thinning slope and the effect of fiber fraction could be well predicted with the model. Meanwhile, the model over-predicts N1 than the experimental data, and this is mainly due to the over-prediction by the original Leonov model for the neat polymer melt. However, the increasing slope and overall shape of the first normal stress difference curve are well predicted by the model. Future experimental works especially with both the steady and transient rheology for the short fiber reinforced thermoplastic melts should be highly worth-


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Fig. 3. (a) Fiber and disk in a simple shear flow and (b) evolution of their orientation vector components.

while for further investigation of the fiber-polymer coupling effects. Also the experimental data for the fiber orientation state should be worth investigating along with the model prediction.

3. Disk Orientation and Surface Color in Injection Molding This section introduces more industry-oriented subject of disk orientation in injection molding (Park et al., 2011). A composite material, in which pigment flakes, e.g. aluminum and mica flakes, are compounded with a thermoplastic resin, is now widely used in the industry to manufacture metal-, pearl- or china-like plastic products. For such flake pigmented thermoplastic composites, the surface appearance of the injection molded part is significantly affected by the orientation state of the pigment flakes or disks (Rawson et al., 1999). Therefore, it would be quite important to understand and predict the orientation behavior of the flakes during the injection molding of aesthetic plastic parts. Particularly the orientation behavior at the weldline is of technical importance since the color defect at the weldline becomes more pronounced due to the pigment flake orientation. 222

Since short fiber and flake (or disk) fillers have been widely used in the composite materials for various applications, the major difference between their orientation kinematics is briefly introduced using Jeffery model with a simple shear flow. In this simple study, the fiber diameter is assumed to be negligible, while the disk is assumed to have negligible thickness. The only difference between the fiber and the disk kinematics according to the Jeffery model is their geometric parameter (λ) which is 1 for fiber (having negligible diameter) and -1 for disk (having negligible thickness). Fig. 3 shows the fiber and disk orientations when they are subjected to a simple shear flow. They have the same orientation of p = (1 ⁄ 2, 1 ⁄ 2, 0) initially (Fig. 3(a)). The transient solution of Jeffery model (Eq. 4) is shown in Fig. 3(b) where the evolution of x- and y-components of the orientation vector are plotted as a function of applied strain (γ). The fiber aligns in the direction of x-axis as strain grows, thus px and py approach monotonically to their steady state value of 1 and 0, respectively. The disk, on the other hand, approaches to a different steady state of p = ( 0, –1, 0) , and one can observe the overshoot of px and significant decrease of py around γ = 1 where the disk orientation vector is nearly parallel to the x-axis. This simple comparison demonstrates the most significant difference between the fiber and the disk orientation kinematics: the disk orientation vector tends to align in perpendicular to the shear direction while the fiber orientation vector tends to align in the direction of shear. Recently, Park et al. (2011) carried out numerical simulation to predict the disk orientation and the surface color in injection molding of pigmented thermoplastic composite. The filling stage of the injection molding process is simulated using Hele-Shaw flow model. Folgar-Tucker model along with a tensor representation of the orientation state (see Eq. 5) is employed to predict the disk orientation using the particle geometrical parameter (λ) value close to -1. Also a phenomenological model is proposed to estimate the surface color from the pigment orientation states. The color intensity (I) is defined as an integration of the orientation tensor component along the thickness direction of the cavity as follows: (8) where ann is the orientation tensor component with the subscript n representing the surface normal direction, and w is a weighting function which can be a function of z, i.e. w = w( z) . At this moment, w is assumed to be constant for a simplicity of the model. Three cavity geometries are studied to examine the effect of the hole and the rib structures on the surface color. Fig. 4 shows the color prediction results in terms of the color intensity distribution. Since the simulation assumes random orientation at the gate, the color intensity is the lowest around the gate region. In the Korea-Australia Rheology Journal


Fig. 5. Color prediction result (top) and real image of injection molded test geometry (bottom).

Fig. 4. Color intensity results for geometries of (a) simple plate, (b) plate with a hole and (c) plate with a rib.

simple plate shown in Fig. 4(a), the color intensity gradually increases along the flow direction as the disks tend to align in normal direction to the plate surface in the shell layer. Around the edge region, the color intensity is relKorea-Australia Rheology Journal

atively low because of the different orientation state due to the in-plane velocity gradient effect. The color intensity is significantly reduced along the weldline as shown in Fig. 4(b). Also one can observe the decrease of the color intensity around the rib structure in Fig. 4(c). This kind of variations of the color intensity would be regarded as defects in the injection molded aesthetic part. Fig. 5 compares qualitatively the prediction result of the color intensity and the real image of the injection molded part. The dark regions in the numerical result indicate the defects such as the weldline and the flow mark. The overall distribution of the predicted defects (such as flow mark due to the rib and weldlines due to the holes) is in a good agreement with the real image of the injection molded part. This result suggests that the proposed modeling and simulation would be useful to predict the possible locations of the color defects. Also experimental studies have been carried out to observe the flake orientation in injection molded part. Fig. 6 shows the transmitted microscopic images of the flake orientation in the injection molded sample shown in Fig. 5(b). One image (Fig. 6(a)) is from the location without any surface color defect and the other (Fig. 6(b)) is from the weldline location. Generally one can observe the sandwich structure where the flake orientation near the midplane (namely the core layer) is significantly different from that around the surface (namely the shell layer). In the shell layer, the shearing dominates the material deformation, which drives the flakes to orient in perpendicular to the surface plane. In the core layer, on the other hand, the shear is relatively small and velocity profile is symmetric with respect to the mid-plane, thus the flakes tend to have inplane directional orientation. At the weldline region, as shown in Fig. 6(b), there is locally different orientation state near the top and bottom surfaces where the flakes orientation is not perpendicular to the surface plane. The different orientation at the weldline is because of the fountain


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Fig. 6. Transmitted microscopic images for specimens (a) without weldline and (b) with weldline.

flow developing at the melt front region: the flakes near the core layer move to the shell layer in the melt front region due to the fountain flow; this will result in the locally different orientation at the weldline where the two fountain flows meet with each other; and the flake orientation becomes frozen-in if the cooling takes place faster than the particle reorientation. Since the orientation significantly affects the light reflection property at the surface of the molded part, the different orientation along the weldline will appear as a color defect in a macroscopic view. According to the numerical and experimental studies, the location and distribution of the surface color defect could be reasonably predicted using the existence models. For more accurate prediction of the orientation and surface color, however, there still needs further studies regarding, for instance, particle interaction, three-dimensional flow effect, the orientation-color relationship, and so on.

4. Rheology for Magnetic PIM Finally, we introduce our recent efforts toward microand precision-magnetic PIM. PIM process is a promising manufacturing technology for a mass production of complex three-dimensional metal and ceramic parts. The process consists of four steps: (i) feedstock preparation (ii) 224

Fig. 7. Effect of magnetic field on (a) the shear viscosity and (b) the first normal stress different for suspension of stainless steel powder (17-4PH) in a wax.

injection molding (iii) debinding and (iv) sintering. In the first step, the metal or ceramic powders are mixed with a binder system commonly by using extruders. Then, injection molding is carried out in a similar manner to the conventional injection molding process but with much lower temperature at the screw and nozzle. The binder system in the injection molded part is removed by chemical or thermal processes. Then the final part is obtained after sintering process. In the magnetic PIM, the orientation of the magnetic particle is the most important factor affecting the magnetic property of final part. In order to obtain a designed specific orientation of the magnetic particles in the injection molded part, one can apply magnetic field in the mold cavity during the injection molding process. As the external magnetic field affects the magnetic particle behavior, the Korea-Australia Rheology Journal


rheological property of the feedstock material becomes more complex, which in turn affects the flow behavior during injection molding process. Therefore, for a systematic control of the magnetic particle orientation, injection molding process and optimal design of the final part, it is of great importance to understand the rheological behavior of the magnetic feedstock. In those regards, we have been carrying out experimental characterization for the rheology of various feedstock materials. For a binder system, only the paraffin wax is used to discard the nonlinear behavior due to other polymer additives. A magnetic powder is mixed with the binder at a volume percentage of 10%. Rotational rheometer with a magnetic device (MCR rheometer, Anton Paar) is used for all experiments. The magnetic field direction is parallel to the rotational axis. Fig. 7 shows the shear viscosity and the first normal stress difference for the suspension with paramagnetic powder of stainless steel (17-4PH) at various magnetic fields. The results show that the rheological properties change significantly as the external magnetic field varies. The shear viscosity and the first normal stress difference increase with the magnetic field, which would be due to the chain formation of the stainless steel particles by the magnetic field. The length and the rigidity of the chain increase with the magnetic field, thus it requires more energy to drive the shear flow as the magnetic field strength increases, which is reflected by the increase of the shear viscosity. Also the chain structure results in the increase of the first normal stress difference as magnetic field increases. More experiments are in progress and the results will be discussed in the future publication. Along with the experimental characterization of the magnetic suspension rheology, also the theoretical and numerical studies are worth investigating for prediction of magnetic particle orientation and optimization of the magnetic PIM process. At this moment, we are attempting to develop a new model and simulation program to predict the magnetic particle orientation during injection molding. Details of the progress in this subject could be discussed later.

5. Summary In this paper, recent progresses and developments in rheology and processing of suspension materials are introduced. Three representative studies for fiber, disk and magnetic particle suspensions are discussed for rheological modeling, orientation behavior and rheological characterization, respectively. Although these subjects are treated independently, they have the same fundamental background which is about (i) the kinematics of the particles and (ii) the stress contribution due to the particles. In the first subject for fiber suspension, a new rheological model, where the fiber anisotropic effect is taken into account in Korea-Australia Rheology Journal

the viscoelastic deformation of the matrix, is introduced. The fiber and polymer kinematics and the stress expression are briefly explained. Further studies regarding the transient rheological behavior are worth investigating along with the experiments using pre-shearing or flow-reversal shear flows. The second subject focuses on the surface color of flake pigmented thermoplastic. The numerical simulation for predictions of the disk orientation and the surface color property are introduced. By using a phenomenological model for orientation-color relationship, the distribution and location of color defects could be reasonably predicted. Also some experimental observations on the flake orientation in the injection molded part are presented. In the final subject, our recent efforts for magnetic PIM have been presented. Overall scope of the micro- and precision-magnetic PIM project is introduced, and some rheological data for magnetic particle suspension are discussed. With the systematic characterization of the rheological behavior of various feedstock materials, the optimization of the process and design for high performance magnetic components would be achieved.

Acknowledgement This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0027626) and by WCU (World Class University) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R31-30005).

List of symbols a A B1, B2 ce C1, C2 C2 D ep h I I1, I2 J1, J2 L nf N1 Np p p px, py δ φf

the second-order orientation tensor the fourth-order orientation tensor functions of joint invariants of a and ce reversible Finger strain tensor functions of invariants of ce fiber interaction coefficient fiber diameter dissipative rate-of-strain tensor average spacing between neighboring fibers color intensity the first and the second invariants of ce the first and the second invariants of a fiber length fiber number density the first normal stress difference particle number particle orientation vector pressure particle orientation vector components identity tensor fiber volume fraction


225


γ· γ·

η ηm ηp ηs λ θ ω

rate-of-strain tensor generalized rate-of-strain shear viscosity matrix viscosity polymer viscosity solvent viscosity particle geometrical parameter polymer relaxation time vorticity tensor

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