Polymer-Particle Filler Systems

2

Polymer-Particle Filler Systems

2.1

Introduction

In this chapter we consider the characteristics of binary polymer-solid particle suspensions. Our concern is with polymer-particle interaction and particle-particle interactions, especially in their roles to influence the melt flow and enhance solid mechanical behavior. We discuss the behavior of isotropic- and anisotropic-shaped particle compounds in thermoplastics, including rheological behavior from low loadings to high loadings obtained using various instruments. Many of the fillers used in industry are anisotropic in character. Depending on the shape of fillers, they are subdivided into isotropic particles, flakes, and fibers. Anisotropic particles may take on states of orientation because of flow and packing processes. Whether developed during flow or processing, particle orientation influences phenomena ranging from rheological properties to compound processability in industrial processing equipment, electrical characteristics, and mechanical performance.

2.2

Particle Properties and Interaction

Depending on the physical properties and size of fillers, the behavior of particle-filled suspensions and filled polymer compounds change. Such properties primarily include particle density, shape, and interaction. To these might be added particle hardness, refractive index, thermal conductivity, electrical conductivity, and magnetic properties.

2.2.1

Particle Density

The densities of particles suspended in liquids and solids range from 0.03 to 20 g/cm3, but most particles have densities that range from 2 to 3 g/cm3, as shown in Table 2.1. The density of a compound ρc is determined by the density of the polymer and the suspended particles. In a uniform particle distribution, the specific volume of a compound (volume per unit mass) may be represented in terms of the volume fractions φpart (particles) and φmatrix (matrix) of its components and their specific volume νpart and νmatrix : νc = φpart νpart + (1 − φpart ) νmatrix or equivalently

(2.1a)

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Table 2.1

Densities of Important Particle Reinforcers Fillers

Particle

Density (g/cm3)

Particle

Density (g/cm3)

Expanded polymeric microspheres Hollow glass beads Thin-wall hollow ceramic spheres Wood flower Porous ceramic spheres Silver coated glass beads Thicker wall hollow ceramic spheres Polyethylene fibers and particles Cellulose fibers Unexpanded polymeric spheres Rubber particles Expanded perlite Anthractite Aramid fibers Carbon black PAN-based carbon fibers Precipitated silica Pitch-based carbon fibers Fumed and fused silica Graphite Sepiolite Diatomaceous earth Fly ash Slate flour PTFE (polytetrafluoroethylene) Calcium hydroxide Silica gel Boron nitride Pumice Attapulgite trihydroxode Magnesium oxide and hydroxide Unexpanded perlite Solid ceramic spheres Solid glass beads Kaolin and calcinated kaolin Silver coated glass spheres and fibers Glass fibers Feldspar

0.03–0.13 0.12–1.1 0.24 0.4–1.35 0.6–1.05 0.6–0.8 0.7–0.8 0.9–0.96 1.0–1.1 1.05–1.2 1.1–1.15 1.2 1.31–1.47 1.44–1.45 1.7–1.9 1.76–1.99 1.9–2.1 1.9–2.25 2.0–2.2 2.0–2.25 2.0–2.3 2.0–2.5 2.1–2.2 2.1–2.7 2.2 2.2–2.35 2.2–2.6 2.25 2.3 2.4 2.4 2.4 2.4–2.5 2.46–2.54 2.5–2.63 2.5–2.8 2.52–2.68 2.55–2.76

Clay Hydrous calcium silicate Vermiculite Quartz and sand Pyrophyllite Aluminum powders and flakes Talc Nickel coated carbon fiber Calcium carbonate Mica Zinc borate Beryllium oxide Dolomite Wollastonite Aluminum borate whiskers Zinc stannate and hydroxy stannate Silver coated aluminum powder Apatite Barium metaborate Titanium dioxide Antimony pentoxide Zinc sulfide Barium sulfate and barite Lithopone Iron oxides Sodium antimonite Molybdenum disulfide Antimony trioxide Zinc oxide Nickel powder or flakes Copper powder Silver coated copper powders + flakes Molybdenum powder Silver powder and flakes Gold powder Tungsten powder

2.6 2.6 2.6 2.65 2.65–2.85 2.7 2.7–2.85 2.7–3.0 2.7–2.9 2.74–3.2 2.8 2.85 2.85 2.85–2.9 2.93 3.0–3.9 3.1 3.1–3.2 3.3 3.3–4.25 3.8 4 4.0–4.9 4.2–4.3 4.5–5.8 4.8 4.8–5.0 5.2–5.67 5.6 8.9 8.92 9.1–9.2 10.2 10.5 18.8 19.35

2.2 Particle Properties and Interaction

φpart 1 − φpart 1 = + ρ ρpart ρmatrix

75

(2.1b)

where the densities ρ are reciprocals of the specific volumes. Compound density in products can vary locally due to uneven distribution of fillers. For example, in one study on glass-filled polystyrene in an injection molding process, the density varied between 0.9 and 1.4 g/cm3, depending on sampling location and process condition in an injection molded bar [1].

2.2.2

Particle Size

The particle sizes of industrially used particulates vary from 50 to 100 μm at the largest level to 0.01 μm (10 nm) on the smallest scale (see Table 2.2). At the largest scale are metallic flakes and powders and some glass beads. The diameters of glass, aramid, cellulosic, and carbon fibers are usually 10 to 12 μm. Mineral fillers, such as calcium carbonate, mica, and talc, are in the range of 0.07 to 5 μm. Titanium dioxide is about 0.1 μm. Carbon black and silica are primarily in the range of 0.01 to 0.05 μm. The smallest mineral particles are silicate layers of fully exfoliated clays (usually montmorillonite). The size of particles is often discussed in terms of surface area. For spherical particles of diameter d, the area per unit mass of particle (a = A / M ) is

a=

Apart M part

⎡ ⎤ ⎢ ⎥ 2 6 πd ⎢ ⎥= =⎢ 3 ⎥ ρpart d ⎛πd ⎞ ⎢ ρpart ⎜ ⎟⎥ 6 ⎝ ⎠ ⎦⎥ ⎣⎢

(2.2)

We may also use d , the average particle diameter, in place of d. This might be defined, based on Eq. 2.2, by d =

∑ N i di3 ∑ N i di2

(2.3)

Different particle size averages may be defined analogously to the averages used for molecular weights (Eq. 2.1). Generally, particle surface areas consist of a sum of both external surface area aext and internal surface area aint with particles. The latter is very important for porous catalyst particles. We may write a = aext + aint

(2.4)

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Table 2.2

Commercial Particle Dimensions

Particle type

Diameter (μm)

BET area (m2/g)

Aluminum borate whiskers Aluminum oxide Aluminum trihydroxide Antimony trioxide Attapulgite Barium Sulfate Barium titanate Bentonite Boron nitride Calcium carbonate Calcium carbonate (precipitated) Calcium hydroxide Carbon black Ceramic beads (hollow) Ceramic beads (solid) Clay Copper Cristobalite (micronized) Cristobalite (coarse) Diatomaceous earth Ferrites Feldspar Glass beads Gold Graphite Hydrous calcium silicate Iron oxide Kaolin Lithopone Magnesium hydroxide Mica Nickel Perlite Sepiolite (hydrated magnesium silicate) Fumed Silica Fused silica Precipitated silica Quartz silica (microcrystalline silica, tripoli, novaculite)

0.5–1 13–105 0.7–55 0.2–3 0.1–20 3–30 0.07–2.7 0.18–1 3–200 0.2–30 0.02–0.4 5 15–250 50–350 1–200 0.4–5 1.5–5 0–6 0–200 3.7–25 0.05–14 3.2–14 7–8 0.8–9 6–96 9 0.8–10 0.2–7.3 0.7 0.5–7.7 4–70 2.2–9 11–37 5–7 5–40 4–28 1–40 2–19

2.5 0.3–325 0.1–12 2–13 120–400 0.4–31 2.4–8.5 0.8–1.8 0.5–25 5–24 – 1–6 10–560 0.1–1.1 19–31 – 0.4–6.5 – 0.7–180 210–6000 0.8–4 0.4–0.8 0.05–0.8 6.5–20 95–180 30–60 8–65 3–5 1–30 1.1–2.6 0.6–0.7 1.88 240–310 50–400 0.8–3.5 12–800 –

2.2 Particle Properties and Interaction

Table 2.2

77

(continued)

Particle type

Diameter (μm)

BET area (m2/g)

Sand (silica flour, ground silica) Silica gel Silver powder and flakes Talc Titanium dioxide Wood flour Wollastonite Walnut shell flour Zeolites Zinc borate Zinc oxide Zinc sulphide

2–90 2–15 0.25–25 1.4–19 8–300 10–100 1–50 9.0 50 0.6–1 0.036–3 0.3–0.35

0.3–6 40–850 0.15–6 2.6–35 7–162 55–60 (oil absorption) 0.4–5 0.5 – 10–15 10–45 8

It is well known to scientists who work with suspensions of particles or polymer-particle compounds that there is a fundamental change in the behavior of particles of these systems as the particle dimensions change. When the particles are large, hydrodynamic effects dominate their flow (rheological) behavior and similar elastic characteristics dominate their modulus and ultimate behavior. When the particle sizes are reduced, the attractive forces between them become increasingly important. The critical dimension distinguishing these two regimes is in the neighborhood of 5 μm but depends upon the particle material. Researchers [2 to 10] who have studied the rheological properties of suspensions of particles in both Newtonian liquids [2 to 4] and polymer fluids [5 to 10] have long found that the shear viscosity increases as the volume fraction of particles increases. Further, the shear viscosity at constant volume fraction solids increases as the particle size decreases. In addition, the rheological responses become more complex and greater time dependences of stress responses are found as particle size decreases. Yield values, a stress below which there is no flow, develop and increase with both higher particle loading and decreasing particle size [2, 4, 10 to 26]. This was first clearly seen in rubber compounds by Scott [11] as early as 1931. The effects are greater when the interparticle forces are stronger.

2.2.3

Particle Shape

Particles have a range of shapes. Some are roughly spherical. Others are aggregates of covalently bonded spheres (carbon black, silica). Still other widely used particles are flakes (mica, talc, clays) and fibers (cellulose, melt spun glass, synthetic fibers). Particles of the latter type are capable of orienting and forming anisotropic compounds where mechanical, optical, and thermal properties vary with direction. Regions of local order develop in normal isotropic systems of fibers or flakes because of difficulties in packing these particles together.

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2.3

Hydrodynamic Theory of Suspensions

2.3.1

General

In the first decade of the 20th century, Einstein [27] gave his attention to the prediction of the viscosity of a dilute suspension of spheres. Einstein began with the creeping flow NavierStokes equation of Newtonian fluid hydrodynamics: 0 = −∇p + η ∇2 ν

(2.5a)

The fluid was considered to adhere to the surface of the sphere ν (R, θ, φ) = 0

(2.5b)

Here ν is velocity, p is pressure, η is viscosity, and R is the radius of the sphere. Stokes [28] had earlier used Eq. 2.5 for analyzing the flow around a sphere. Einstein argued that the viscosity of the suspension could be determined by the increase of the energy dissipated, Φ, by the presence of the sphere, i.e., η Φ = = η0 Φ0

∫ σ ⋅ ν ⋅ n d a

∫ σ

(0)

⋅ ν(0) ⋅ n d a

(2.6)

where σ is the stress tensor, n is a unit normal to the surface ‘a’ of the suspended sphere; η 0 is the viscosity of the suspending medium and σ(0) and ν(0) the stress and velocity fields in the absence of the sphere. He was able to show that the viscosity, at very low volume fractions of spheres, increased according to η = 1 + 2.5 φ η0

(2.7)

where φ is the volume fraction of the spheres. Eq. 2.7 is valid only for extremely dilute suspensions, in which interactions between neighboring particles are negligible (i.e., in the absence of hydrodynamic interactions). The method of Eq. 2.6, however, can also be applied to more concentrated systems of suspended particles. In 1945 Guth [29] noted the similarity of the mathematical formulations of the theory of linear elasticity with those of Newtonian fluid hydrodynamics. He saw that if the elastic solid were incompressible that the presence of rigid spheres would similarly increase the modulus G according to elastic energy arguments, specifically, G W = = G0 W0

∫ σ ⋅ ε ⋅ n d a

∫ σ

(0)

⋅ ε(0) ⋅ n d a

(2.8)

2.3 Hydrodynamic Theory of Suspensions

79

where W is the elastic strain energy, ε is the infinitesimal strain and G0 is the modulus of the suspending medium and σ(0) and ε(0), the stress and strain field in the absence of the sphere. This leads to for dilute systems of spheres to G = 1 + 2.5 φ G0

(2.9)

which is the same as Einstein’s Eq. 2.7.

2.3.2

Spheres

The formulation described above is found to be valid for large particles about 10 μm or greater. It does not consider van der Waals and dipole-dipole forces between particles. It is, as we will see, able to consider concentration effects and the influence of particle anisotropy. The effect of concentration on viscosity of suspensions of spheres was first modeled by Guth and Simha [30]. They again began with Eq. 2.5, but considered the hydrodynamic interaction of other spheres on the velocity field. This led to an enhancement of the viscous dissipation and to an increase in suspension viscosity of form η = 1 + 2.5 φ + C φ2 η0

(2.10)

The value of C has been variously given as 14.1 and 12.6 [30, 31]. It follows from the previously mentioned arguments of Guth [29] that the elastic modulus for an incompressible solid should be equivalently given by G = 1 + 2.5 φ + C φ2 G0

(2.11)

Studies of the viscosity of suspensions at higher concentrations have usually used cell models. Here, one essentially considers the spheres are arranged in a lattice-like array, and one analyzes the flow in a representative cell. Cell models are critically discussed in the monograph of Happel and Brenner [32]. Such an approach was initiated by Simha [33]. He analyzed the flow around a sphere located in a cell. He predicted η(φ) = 1 + 5.5 Ψ(φ) ⋅ φ η0

(2.12)

where φ is volume fraction and Ψ(φ) is a concentration dependent interaction parameter. When ν is small, Ψ(φ) is equal to unity. Thus Simha’s formulation does not reduce to Einstein’s Eq. 2.7. Variant cell theories of concentrated suspension of spheres in Newtonian fluids have been developed by Happel [34], and Frankel and Acrivos [35].

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Tanaka and White [36] using the Frankel-Acrivos cell theory formulation have modeled the shear flow of a concentrated suspension of spheres in a power law non-Newtonian fluid (Eq. 2.5b). Mooney [37] has developed a formulation of the viscosity of suspensions of spheres based on a functional equation, which must be satisfied if the final viscosity is to be independent of the sequence of stepwise additions of partial volume fractions of the spheres to the suspension. Further Einstein’s Eq. 2.7 is considered valid for dilute suspensions, Mooney argues for a monodisperse system of spheres that ⎡ 2.5 φ ⎤ η = exp ⎢ ⎥ η0 ⎣1 − k φ ⎦

(2.13)

where k is a self-crowding factor. The parameter k was found experimentally to be between 1.35 and 1.91.

2.3.3

Ellipsoids

In 1922 G. B. Jeffery [38] sought to generalize Einstein’s analysis for a dilute suspension of spheres to ellipsoids. Like Einstein [27], he considered the sphere to be in shear flow (0) ν1(0) = γ x2 , ν(0) 2 = ν3 = 0 . The flow creates stresses at the surface, which causes rotation as well as a translation of the ellipsoid. During shear flow, the particles undergo motions involving regular orbits, which have become known as Jeffery orbits. The long axes of prolate and oblate particles are immersed in a fluid in laminar motion. They will tend to set themselves parallel to the flow direction. Using the creeping flow Navier-Stokes equations, he found the stresses at the ellipsoid surface are equivalent to the sum of two couples. One tends to make an elongated particle (prolate ellipsoid of revolution) adopt the same rotation as the undisturbed flow. This makes the particle axis tend to lie parallel to the 3 direction and adopt the same rotation as the flow field. The other tends to make an elongated particle rotate in the 1-2 plane. Jeffery [38] predicted that the viscosity of a very dilute suspension of ellipsoids in a shear field of Newtonian liquid would have the form η =1+ ν φ η0

(2.14)

where ν is a parameter which depends on the aspect ratio and orientation of the ellipsoidal particles. For spheres it is 2.5. Jeffery found that ν varies from 2.0 to very large values for prolate spheroids, which approach rods. A major significance of Jeffery’s studies lies in the derivation of equations of anisotropic particle motion in a Newtonian liquid. He predicted that shear flow rotates the disc/rod in shear planes. Jeffery hypothesized that in time the orbital motions would evolve to the one that corresponds to the least viscous dissipation. The motion of single ellipsoids in the shear

2.3 Hydrodynamic Theory of Suspensions

81

flow of water has been investigated experimentally by Taylor [39, 40]. Particle orbits in the flow of dilute suspensions of ellipsoids have also been observed by Mason and coworkers [41 to 44]. We discuss this in Section 2.4.2.1. There are no equivalent studies of concentrated anisotropic suspensions. This is because of the tendency of anisotropic particles to form oriented ordered structures such as “micro-bundles” of fibers. Batchelor [45] has modeled the flow of concentrated suspensions of large parallel fibers. Very large elongational viscosities are predicted. This was extended to suspensions in non-Newtonian fluids by Goddard [46].

2.3.4

Interacting Particles

We may modify the above approaches to include interparticle forces. It is possible in this way to predict the non-Newtonian responses due to interparticle forces and yield values. These may be incorporated into cell models of suspensions. An early study of this was by Tanaka and White [36] using a power law fluid matrix.

2.3.5

Continuum Theories

Continuum mechanics theories of various types have been developed through the years to represent the flow of suspensions. The earliest efforts of this type were one-dimensional, but from the 1940s there have been three-dimensional efforts, which we will emphasize here. Scott [11] in a 1931 study, one of the earliest studies of the rheology of rubber compounds for these materials, suggested combining a yield value, Y, and a power law σ = Y + K γ n

(2.15)

for shear flow. Here σ is a shear stress, n the power law exponent, and K, a coefficient. Expressions of this type were widely used in succeeding years to model the flow of suspensions. In papers published in 1947 and 1948, Oldroyd [47] developed a three-dimensional of Eq. 2.15 that could predict non-Newtonian viscosity and three-dimensional “plastic” yielding based on a von Mises stress criterion. Subsequently, Slibar and Pasley [48] developed a similar three-dimensional model, which accounted for thixotropy. In 1979 White [49] proposed a three-dimensional theory of a plastic-viscoelastic fluid intended to represent the behavior of small particle-filled compounds. This was also based on the von Mises stress criterion but contained memory and predicted shear flow normal stresses. This was later generalized to include thixotropy [50, 51]. Leonov [52] developed an alternative three-dimensional tensor theory of this behavior. In more recent years, White and Suh [53] and White et al. [54] have developed threedimensional models of compounds with anisotropic disklike particles, which exhibit yield values. These produce direction dependent flow characteristics. Later Robinson et al. [55] described a transversely isotropic Bingham fluid model.

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2.4

Experimental Studies of Compound Properties

2.4.1

Large Spheres

The viscosities of suspensions of large non-interactive spheres have been made by various investigators [56 to 60]. The agreement of experimental data with the hydrodynamic theory seems not clear in either dilute or concentrated systems. The best correlation of data with a model over wide range concentrations is with Mooney’s [36] Eq. 2.13. At very high concentrations, Reynolds [61] called attention to the possibility of a phenomenon he called “dilatancy” occurring. By this he meant that shear flow would induce an expansion of the suspension during flow. This is caused by the break up the lattice like structure, which might develop at rest in a concentrated suspension of spheres. Subsequently, Freundlich and Roder [62] presented arguments in 1938 that Reynolds dilatancy would involve not only a dilation of the suspension but also an increase in the shear viscosity. In subsequent years, various investigators searched for the Reynolds [61] and FreundlichRoder [62] effects in concentrated suspensions of small particles. Of special interest are the researches of Metzner and Whitlock [63] and Hoffmann [64]. The former authors [63] described experiments on a titanium dioxide particle suspension of unspecified particle size and suspensions of large glass spheres (28 to 100 μm). Both volumetric dilation and Freundlich-Roder viscosity increases were only observed in the titanium dioxide suspensions, but appeared to occur independently with volumetric dilation occurring at lower shear stresses/rates.

Figure 2.1 Hoffman’s [64] study of the viscosity of suspension of polyvinyl chloride particles

2.4 Experimental Studies of Compound Properties

83

Hoffman [64] studied monodisperse suspensions of polyvinyl chloride and styrene-acrylonitrile copolymer particles of diameter 0.4 to 1.25 μm using both rheological and structural characterization techniques as shown in Figure 2.1. The shear viscosities showed striking viscosity increases at critical shear rates. Structurally, the suspensions at low shear rates were found to exhibit a hexagonal crystalline lattice. At a critical shear rate, this lattice structure broke up into less-oriented arrays with a jump increase in shear viscosity. There are few studies of suspensions of large particles in polymer melts. Nazem and Hill [65] investigated the influence of glass spheres of diameter 10 to 60 μm on melt viscosity in a styrene-acrylonitrile copolymer (SAN) matrix. Generally, the viscosity is increased by the presence of the spheres. The general shape of the viscosity-shear rate curves was unchanged.

2.4.2

Large Fibers and Ellipsoids

2.4.2.1

Fluids

As noted earlier, in the 1920s Taylor [38] experimentally verified Jeffery’s [37] analysis for the motion of ellipsoids. There were subsequent studies by Taylor in the 1930s [39, 40]. In the 1950s Mason and his coworkers [41 to 44] made extensive efforts to visualize anisotropic particle motions in dilute suspensions during flow of rigid rod- and disk-shaped particles. They observed a distribution of orbits. There is a significant literature beginning in the 1970s [66 to 73] on the flow of glass fiber suspensions in both Newtonian fluids and in polymer melt matrices. This literature describes rheological properties including the shear viscosity, normal stresses, and elongational flow characteristics. It also includes the development of fiber orientation in flow and fiber breakage. These fibers were of order 10 to 12 μm in diameter, and there are only very weak attraction forces. Maschmeyer and Hill [69] describe experimental studies of glass fibers suspended in silicone oil. There were great problems associated with reproducibility, fiber breakage, and clogging of capillary die entries. Most striking was their observation of Weissenberg normal stress effects (i.e., rod climbing by the suspension when stirred). Normal stress effects, as discussed in Section 1.3.5, are commonly found in polymer solutions. It is surprising at first, but on later consideration instructive, to see them in fiber suspensions in a Newtonian liquid. The first extensive experimental investigation on the rheological properties of glass fiberpolymer melt suspensions was by Chan et al. [70 to 72]. They investigated polymer melt compounds of polyethylene, polystyrene, and isotactic polypropylene. These authors found that the fibers enhanced the shear viscosity of melts, but did not induce yield values (see Fig. 2.2). The compound exhibited very large normal stresses and a high elongational viscosity (see Fig. 2.3). A subsequent paper by Czarnecki and White [73] confirmed these observations on the shear viscosity and normal stresses in polystyrene compounds. This paper also makes comparisons with aramid and cellulosic fibers, whose suspensions in polymer melts have similar behavior. They showed that masticating glass fiber compounds damages and breaks fibers. It was found that this reduces the normal stresses (Fig. 2.4).

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2 Polymer-Particle Filler Systems

Figure 2.2 Elongational viscosity of a suspension of a suspension of glass fiber (φ = 0.2 () and 0.4 ()) in a polystyrene melt (Chan et al. [70])

Figure 2.3 Principal normal stress difference N1 vs. shear stress with 0 (), 10 (), and 22 () vol% glass fibers (Chan et al. [70])

2.4 Experimental Studies of Compound Properties

85

Figure 2.4 Principal normal stress difference N1, as function of shear stress σ12 for a polystyrene melt and 22 () volume percent glass fibers before mastication and after 30 and 60 min mastication (Czarnecki and White [73])

O’Connor [74] was the first to compare breakage of different reinforcing fibers during mixing. Czarnecki and White [73] compared the breakage of aramid and cellulose fibers with glass fibers. Glass fibers become severely damaged rapidly. Both aramid and especially cellulose fibers break more slowly. The aramid fibers broke into a segmented structure. The cellulose fibers are annular cylinders that have collapsed and buckled (Section 1.4.12.1). With their resulting helical shape, they are much more flexible and did not similarly break. Extensive studies of glass fiber damage in continuous mixers have in more recent years been described by Franzen et al. [75] and by Shon and White [76]. Forgacs and Mason [77] and later Czarnecki and White [73] investigated the mechanisms of fiber breakage during flow and concluded that it was associated with Euler buckling of the filaments in shear flows. 2.4.2.2

Solids

High modulus continuous filament fibers significantly enhance the Young’s modulus and tensile strength of polymers. This is illustrated by a simple derivation. If the fibers are all

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2 Polymer-Particle Filler Systems

parallel, the strain of the fibers and matrix must be the same in a mechanical test. The applied force is F = [ Af Ef + Am Em ] γ = Ac EcII γ

(2.16a)

where Ef and Em are the modulus of fiber and matrix and Af and Am are the cross-section of fiber and matrix, EcII is the composite modulus parallel to the fibers. This leads to the composite modulus [78], EcII = φf Ef + (1 − φf ) Em

(2.16b)

where φf is the volume fraction of fibers taken as Af / Ac . Clearly, Ef quickly overwhelms Em as φf increases. The above behavior is not found when the continuous filaments are perpendicular to the direction of stretch. Consider the perpendicular modulus E⊥ here the deformation of matrix and fibers are additive. Let δ lc be the deformation of the composite, δ lf that of the fibers and δ lm that of the matrix. δ lc = φf δ lf + φm δ lm

(2.17a)

This lead to [78] φ φ 1 = f + m ⊥ E Em Ec f

(2.17b)

The orientation of chopped fibers developed during flow/processing can strongly influence various properties of compounds, such as modulus and tensile strength. In 1974 Brody and Ward [79] described fiber orientation in uniaxial fiber filled composites using Hermans orientation factors [80 to 82] originally developed to measure polymer chain orientation in man-made fibers (see Fig. 2.5(a)). f =

3 cos2 φ − 1 2

(2.18)

where φ is the angle between the fiber axis and the preferred symmetry direction. Subsequently White and Knutssen [83] generalized these considerations to biaxial orientation of fibers. They used the biaxial orientation factors [84] f1B = 2 cos2 φ1 + cos2 φ2 − 1

(2.19a)

= 2 cos φ2 + cos φ1 − 1

(2.19b)

f 2B

2

2

where 1 and 2 now represent two orthogonal directions and angles φ1 and φ2 are the angles between these directions (see Fig. 2.5(b)). These orientation factors were originally developed by White and Spruiell [84] to represent biaxial molecular orientation in polymers. Eqs. 2.18 and 2.19a,b are equal to (0,0) for isotropic random orientation. For perfect uniaxial orientation they are equal to (1,0) or (0,1). For equal biaxial orientation in a plane they are (1/2,1/2).

2.4 Experimental Studies of Compound Properties

preferred direction

1

fiber





 2

3 f1B = 2cos2 φ1 + cos2 φ2 − 1

3 cos2 φ − 1 f = 2

f2B = 2cos2 φ2 + cos2 φ1 − 1

(a) Uniaxial orientation

(b) Biaxial orientation

f1B Uniaxial (Flow Direction) (1,0)

Plannar (Flow Direction and Perpendicular to the Surface)

Plannar (Film Surface) Equal Biaxial Uniaxial (Transverse Direction) (0,1)

Isotropic (-1,-1) (c) White and Spruiell orientation triangle Figure 2.5 Definitions of orientation factors and key angles [84]

f2B

87

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2 Polymer-Particle Filler Systems

In more recent years, Advani and Tucker [85] have independently developed a similar representation of chopped fiber orientation. X-ray diffraction was first used to detect fiber orientation in polymer compounds by Schierding [86] and subsequently by Yoshida et al. [87] and by Menendez and White [88]. This technique was later applied by Lim et al. [89, 90] to study fiber orientation development in processing short fiber–filled compounds. Aramid fibers, which are highly crystalline and oriented, as well as giving sharp diffraction peaks, were used by the latter authors [89, 90]. Generally, it is found that fibers orient with their axis parallel to the direction of flow and perpendicular to the direction of shear. The level of modulus increase by chopped high modulus fibers is less than for continuous filaments at the same volume loading. There is both a fiber length effect and a fiber orientation effect both of which reduce the modulus. One may account for the fiber length effect by writing Eq. 2.16b as [78] Ec = κ Ef φf + Em φm

(2.20)

where κ is approximately [78, 91] 1−

tanh a x ax

(2.21a)

where a is the aspect ratio l / d and x is the dimensionless factor ⎡ ⎤ ⎢ 2 Gm ⎥ x=⎢ Δ⎥ ⎢ Ef ln ⎥ d⎦ ⎣

(2.21b)

where Gm is the shear modulus of the matrix and Δ is the separation distance between the nearest fibers. The question of the influence of chopped fiber orientation on modulus is more complex. From the view of continuum mechanics, the presence of fibers introduces matrix anisotropy. The modulus becomes a fourth order tensor Gijkl, which correlates stress and strain. The stress σij is related to (infinitesimal) strain γkl by σ ij =

∑∑ Gijkl γkl k

(2.22)

k

The application of Eq. 2.22 and the determination of the Gijkl is discussed by Ward and Hadley [78] among others.

2.4 Experimental Studies of Compound Properties

2.4.3

Carbon Black

2.4.3.1

Thermoplastic Melts/Uncured Compounds

89

Carbon black is one of the most widely found particle reinforcements used in industry, especially in rubber products like tires. Carbon black particles used commercially are submicron in size and can be as small as 0.01 μm (10 nm), as described in Section 1.4.2. They are also primarily used at high loadings. The properties of carbon black compounds are because of the small sizes of the particles and the high loadings strongly influenced by particle-particle interaction.

(a) Figure 2.6 Influence of carbon black loading on the shear viscosity of polymer melt; (a) SBR/N326, (b) EPDM/N220

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2 Polymer-Particle Filler Systems

(b) Figure 2.6 Influence of carbon black loading on the shear viscosity of polymer melt; (a) SBR/N326, (b) EPDM/N220

Generally, the shear viscosity of carbon black filled thermoplastic and rubber compounds increases with particle loading and decreasing particle size [4 to 7, 10, 11, 13, 15 to 18]. The ultimate particle sizes of commercial carbon black particles are in the range of 0.01 to 0.1 μm. In this size range, the interaction between the aggregate particles leads to apparent yield values (i.e., a stress below which there is no flow). Figs. 2.6(a) and 2.6(b) show viscosity-shear stress plots indicating this. The details of the appearance of yield values differ depending upon how low the shear stress range where stresses are measured. Osanaiye et al. [21] measured the true yield values with a sandwich rheometer in the creep mode for rubber-carbon black compounds. They found yield values of order between 560 Pa (EPDM/CB, 80/20 wt%) to 5,500 Pa (EPDM/CB, 50/50 wt%). Similar measurements were made by Li and White [22]. Yield values seem also to exist in elongational flow [15, 16, 25]. Carbon black compounds exhibit strong thixotropic behavior. The classic studies of this behavior are by Mullins [92] and by Mullins and Whorlow [93], which involve both simple compounds and vulcanizates. In the former studies, they describe how rubber-carbon black compounds will “stiffen” on sitting and soften on mastication/working. These studies were done in a rotational rheometer with biconical rotors. The longer the rest period, the greater the torque overshoot and the greater the period of overshoot at the beginning of flow in the rheometer. In a subsequent study on natural rubber and SBR, Montes et al. [10] saw many of the same effects (see Fig. 2.7). They also look at programmed step histories, and steady shearing flow. Storage increases thixotropic effects especially with decreasing particle size.

2.4 Experimental Studies of Compound Properties

91

Figure 2.7 Transient viscosity development in carbon black natural rubber compound after various periods of rest in the viscometer

2.4.3.2

Rubber Vulcanizates

There have been many studies of the influence of carbon black on the properties of rubber vulcanizates. Many of these concern the Young’s modulus and its variation with concentration. As we described in Section 2.3.1, Guth [29] showed that the hydrodynamic theory of Einstein [27] and Guth and Simha [30] could be applied to small strains of elastic solids. However, Guth found that when this theory was compared to experimental data on rubber carbon black vulcanizates, there seemed only agreement at the lowest concentrations. The experimentally determined modulus was higher than the predictions at higher concentrations and increasingly deviated to higher values at smaller particle sizes [29, 94, 95]. The increase of modulus with decrease in carbon black particle size is well known in the rubber industry (Fig. 2.8(a)). Not only is the Young’s modulus enhanced by small particle size in carbon black compounds but so is the tensile strength [95]. Important experimental studies of the mechanical properties of rubber-carbon black vulcanizates were reported by Mullins [91, 94]. He studied the engineering stress (F/Ao)-strain curves of elastomer compounds. Unlike the gum elastomers, the compounds showed a softening (modulus reduction) behavior if successively stretched (Fig. 2.8(b)). Most of the softening occurs in the first deformation. After a few stretching cycles, a steady state is reached. The samples subsequently gradually stiffened on standing. This behavior is often called the “Mullins effect”. Another interesting and related mechanical effect was found by Payne [96]. He observed that the dynamic storage modulus G′(ω) (see Eq. 1.42) of rubber-carbon black compounds depends upon the strain amplitude of an oscillatory deformation, decreasing with increasing strain level (Fig. 2.9).

92

2 Polymer-Particle Filler Systems

26

Carbon black 50phr in SBR 24

Tensile Strength (MPa)

22 20 18 16 14 12 10 8 20

(a)

24

28

39

49

70

180

250

Particle Size (nm)

(b) Figure 2.8 (a) Influence of carbon black particle size on tensile properties of rubber vulcanizates [95] (b) Stress-strain softening behavior (Mullins effect) of carbon black filled natural rubber compound

Generally, the small particle size effects observed with carbon black in compounds result in very high moduli or viscosities as well as thixotropy. These seem associated with particleparticle interactions. This would be seen to be associated with networks of carbon black aggregates, which gradually form under quiescent condition. These break up during severe deformations but can subsequently reform.

2.4 Experimental Studies of Compound Properties

93

Figure 2.9 Influence of strain on the dynamic storage modulus of butyl rubber-carbon black compounds up to 23.2 vol% [96]

2.4.4

Calcium Carbonate (CaCO3)

2.4.4.1

Thermoplastic Melts

Calcium carbonate is the most widely used particle reinforcement in the thermoplastics industry. CaCO3 particles are available in a large range of particle sizes. There have been several investigations of the rheological properties of CaCO3 melt compounds [8, 9, 16, 22, 25]. As with carbon black, the viscosity of polymer melts such as polyolefins and polystyrene is increased by the presence of CaCO3 particles. The higher the particle loading and the smaller the particle size, the greater the viscosity increases (Fig. 2.10). Generally, compared to carbon black, the levels of viscosity increase tend to be higher for particles of roughly the same size [22, 97]. This would seem to be due to greater interparticle forces.

94

2 Polymer-Particle Filler Systems

1011 PS PS / CALCITE PS / CALCITE PS / CALCITE PS / CALCITE

1010

VISCOSITY (Pa.s)

10

9

(95 : 05) (90 : 10) (80 : 20) (60 : 40 )

108 107 106 105 104 103

sandwich rheometer cone-plate rheometer capillary rheometer

102 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104

(a)

SHEAR RATE (sec-1)

(b) Figure 2.10

Influence of calcium carbonate particles on polymer melt viscosity; (a) particle loading [26], (b) particle size at 30 vol% loading [9]

2.4 Experimental Studies of Compound Properties

95

For particle volume fractions of 0.15 and more at low shear rates and shear stresses, the shear viscosity becomes increasingly non-Newtonian and yield values develop [8, 9, 16, 22, 25]. Calcium carbonate acts as a nucleating agent for various thermoplastic melts such as polypropylene. 2.4.4.2

Solid Thermoplastics

There seem to be only limited published studies of calcium carbonate’s influence on the mechanical properties of thermoplastics and elastomers. Clearly, the Young’s modulus and tensile strength are increased. H. Kim et al. [98] showed the addition of calcium carbonate into polypropylene, poly(propylene-random ethylene) copolymer, and poly(propylene-ethylenebutene) terpolymer increases Young’s modulus and decreases elongation to break. Kwon et al. [99] observed the same trend for high-density polyethylene, low-density polypropylene, and linear low-density polyethylene compounds that tensile stress of calcium carbonate–filled compounds were higher than unfilled systems. Suetsugu [100] studied the tensile strength and the impact strength of polypropylene-calcium carbonate compounds and found it to be dominated by the dispersion of the agglomerate. He defined a dispersion index by dispersion index = 1 −

π ∑ Di2 ni 4aφ

(2.23)

where a is the area observed, Di is agglomerate size and ni is the number of agglomerates of size Di. Large agglomerates are found to deteriorate tensile and impact strength (see Fig. 2.11). Generally, large agglomerates are associated with small particles and poor mixing. As described in Section 7.2, stearic acid is often added commercially to calcium carbonate compounds to break up agglomerates and reduce viscosity. It presumably increases tensile strength and impact strength.

Figure 2.11

Influence of calcium carbonate particles on tensile strength of polypropylene compounds [100]

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2.4.5

Silica

2.4.5.1

Uncured Rubber Compounds

Silica particles are commercially available in the same size range as carbon black (i.e., down to 0.01 μm or 10 nm). Silica has been commercial reinforcing filler since the 1940s. The processing of silica-filled rubber compounds is more difficult than with carbon black–filled compounds because polar interparticle forces make it more difficult to break up agglomerates [101]. The viscosity levels of silica-filled compounds are higher at the same volume loading and particle size [102] than carbon black. The viscosity levels are also higher than larger polar particles, such as calcium carbonate, talc, and zinc oxide [102]. Silica-rubber compounds appear to exhibit yield values, higher than other particulates at the same volume fraction. This seems associated with small particle size coupled with strong particle-particle interaction. 2.4.5.2

Rubber Vulcanizates

Michelin has introduced silica as a replacement for carbon black in tire treads [103]. With the introduction of silane coupling agents (Sections 1.7 and 8.3.2), the mechanical properties are enhanced and rolling resistance is lowered. There are, however, no published studies of the detailed mechanical properties of silica compound vulcanizates resembling those on carbon black compounds.

2.4.6

Talc and Mica

2.4.6.1

Thermoplastic Melts

Like calcium carbonate, talc and mica are important mineral fillers. As described in Section 1.4.6, they are structurally layered silicates. Talc and mica are marketed commercially as flakelike particles and are dominated by their layered structures. There are various characteristic features of talc and mica compounds. Among these is the orientability of these flakelike particles. During flow and in processing operations, this behavior of talc and mica has been studied notably by Lim et al. [88, 89], Suh and White [104], and K. J. Kim and White [23]. X-ray diffraction and scanning electron microscopy were used. For slit dies the flakes orient parallel to the die surface. For cylindrical dies, different flow regimes are found [23]. At low volume, loadings the disk surfaces were parallel to the capillary walls, while at higher loadings radial orientation of the disks appears (Fig. 2.12). Studies of the shear viscosity of talc compounds [12, 23, 25, 26] indicate that these compounds behave similarly to carbon black [24] and calcium carbonate [25]. Yield values are induced by volume loadings of 0.15 and higher (Fig. 2.13). Early studies determined yield values by extrapolation. The Osanaiye et al. sandwich rheometer [21], later upgraded by Kim and

2.4 Experimental Studies of Compound Properties

(a)

(b)

(i) Slit dies (a) PS/talc (5 vol%), (b) PS/talc (40 vol%)

(a)

(b)

(c)

(d)

(ii) Cylindrical dies PS/talc (a) 5 vol%, (b) 10 vol%, (c) 20 vol%, and (d) 40 vol% Figure 2.12

Orientation of talc particles developed during melt processing [23]

97

98

2 Polymer-Particle Filler Systems

1011 1010

PS

109

PS / TALC (95 : 05) PS / TALC (90 : 10)

VISCOSITY(Pa.S)

108

PS / TALC (80 : 20) PS / TALC (60 : 40)

107 106 105 104 103 102 101

sandwich rheometer cone-plate rheometer capillary rheometer

100 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 -1

SHEAR RATE (sec )

Figure 2.13

Influence of talc on polystyrene melt viscosity [26]

White [25], measures talc-, talc/calcite-, and calcite-filled thermoplastic compounds [25]. They found the true yield stresses generally give much smaller values than the apparent yield values [21, 22, 24]. Kim and White [25] also found that yield values exist in elongational flow when shear flow yield values exist. 2.4.6.2

Solid Polymers

It is well known that addition of talc and mica to thermoplastics increases Young’s modulus and tensile strength but reduces the elongation to break, as with calcium carbonate [98, 99]. Minerals such as talc, mica, and calcite can act as a nucleating agent for various crystallizing polymers, notably polypropylene [105 to 108]. Specific minerals can induce alternate crystalline forms, such as the polypropylene β structure, in place of the normal α [109]. This can in turn influence the mechanical behavior.

2.4.7

Montmorillonite and Organo Clays (Nano Composites)

Compounds consisting of montmorillonite clay with an organic amine and a polymer are really ternary systems and should be discussed in Section 7.4, where we will consider them in more detail. We will, however, briefly discuss them here. As discussed in Sections 1.4.7 and 1.4.12.1, silicate minerals possess galleries between the silicate layers containing electrical charges. Some of the silicate mineral crystals contain MgO or other compounds between the SiO3 layers and are impenetrable (as is the case of talc).

2.4 Experimental Studies of Compound Properties

99

Montmorillonite clay, on the other hand, can absorb indefinite amounts of H2O into these galleries and displace the SiO3 layers to distances much greater than the initial 0.001 μm or 1 nm [110]. Smith [111] found that if organic amines were introduced into the galleries, it was possible to have exchange reactions with the metallic cations in the clay. The organic cations are held more firmly than the inorganic positive ions. Clay companies subsequently began manufacturing organic amine montmorillonite clays to act as absorbents, thickeners, and in other applications. There have been many subsequent studies of sorption of organic compounds by these clays [112 to 115]. In the 1960s there were studies of absorption of monomers by clays [116, 117]. In more recent years, monomers such as caprolactam were absorbed into organo clays and were subsequently polymerized. The most important of these efforts involved Toyota researchers [118 to 125] using caprolactam. These products were called nanocomposites and seem to contain polyamide intercalated between silicate layers or had exfoliated structures. Other monomers have also been absorbed into montmorillonite and polymerized. These include the formation of polyimides [126], epoxy resins [127], and polylactones [128]. Kojima et al. [122] report that the polyamide-6/montmorillonite compound (nanocomposite or hybrid) has superior modulus, tensile strength, flexural modulus, and impact strength compared to neat polyamide-6, as well as higher heat distortion temperature. Indeed, with 4.7 wt% montmorillonite, the heat distortion temperature was increased 87 °C over neat polyamide-6 to 152 °C. Subsequently various polymers have been successfully melt mixed with into montmorillonite clay [129 to 133] to form nanocomposites. The ability to form such clay nanocomposites seems to depend upon the polarity of the polymers [133] and requires as described above organoamine clay modifier.

2.4.8

Carbon Nano-Tubes

Carbon nanotubes significantly increase the mechanical properties of polymer matrices into which they are introduced. Baughman et al. [134] showed addition of single-wall carbon nanotube into polymer exhibit a high tensile strength and modulus. Single-wall carbon nanotube reinforced filled pitch-based carbon fibers shows better mechanical properties than unmodified pitch-based fibers [135 to 137]. Kumar et al. [138] showed single-wall carbon nanotube filled (10 wt%) poly(p-phenylene benzobisoxazole) (PBO) fiber compound increased tensile strength by 50% compared to unfilled fiber (PBO). Polyacrylonitrile (PAN)/single-wall carbon nanotube (10 wt%) compound increased tensile modulus by 100% [139]. Park and Siochi [140] showed single-wall nanotube-polyamide nanocomposite were electrically conductive (antistatic) and optically transparent with enhanced conductivity by 10 times compared to at 0.1 vol% loading.

100

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2.5

Summary

The addition of small particles to polymers enhances their viscosity in the molten state and their modulus in the solid state. Generally, tensile strength and brittleness are also enhanced. Particles with fibrous or disklike shapes induce anisotropic mechanical and electrical behavior. Suspensions of large particles, ca 10 μm and more, can be modeled by hydrodynamics and the theory of elasticity. With anisotropic particles such as fibers, striking anisotropic characteristics develop. Suspensions with very small particles are quite different in that interparticle forces dominate the rheological and mechanical properties of the compounds. Viscosities and moduli become increasingly large with decreasing particle size. This is especially the case for polar particles. One may also see particle size effects in carbon black compounds when the particles are nonpolar. Yield values (i.e., stress below which there is no flow) develops in polymer melt compounds and ranges of thixotropic effects are found. Effects are seen in solid as well as melt compounds. For polar particles such as calcium carbonate, silica, and zinc oxide, the effects of decreasing particle size on enhancing viscosity are much stronger. These compounds become intractable and additives must be found to modify these properties. This is the subject of Chapter 7.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

S. O. Ogadhoh and T. D. Papathanasiou, Composites Part A: Appl. Sci. Manuf., 27A(1), 57 (1996) E. L. McMillen, J. Rheology, 3, 179 (1932) H. Freundlich and F. Juliusberger, Trans. Faraday Soc., 30, 333 (1934) H. Freundlich and A. D. Jones, J. Phys. Chem., 40, 1217 (1936) J. W. Snyder and M. H. Leonard in Introduction to Rubber Technology, M. Morton (Ed.) Reinhold, New York (1959) C. C. McCabe and W. N. Mueller, Trans. Soc. Rheology, 5, 321 (1961) J. L. White and J. W. Crowder, J. Appl. Polym. Sci., 18, 1013 (1974) T. Kataoka, T. Kitano, Y. Oyanagi, and M. Sasahara, Rheol. Acta, 18, 635 (1975); T. Kataoka, T. Kitano, M. Sasahara, and K. Nishijima, Rheol. Acta, 17, 149 (1978) Y. Suetsugu and J. L. White, J. Appl. Polym. Sci., 28, 1481 (1983) S. Montes, J. L. White, and N. Nakajima, J. Non-Newtonian Fluid Mech., 28, 183 (1988) J. R. Scott, Trans. Inst. Rubber Ind., 7, 169 (1931) F. M. Chapman and T. S. Lee, Soc. Plastics Eng. J., 26(1), 37 (1970) G. V. Vinogradov, A. Y. Malkin, E. P. Plotnikova, O. Y. Sabsai, and N. E. Nikolayeva, Int. J. Polym. Mat., 2, 1 (1972)

References

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.

101

N. Minagawa and J. L. White, J. Appl. Polym. Sci., 20, 501 (1976) V. M. Lobe and J. L. White, Polym. Eng. Sci., 19, 617 (1979) H. Tanaka and J. L. White, Polym. Eng. Sci., 20, 949 (1980) S. Montes and J. L. White, Rubber Chem. Technol., 55, 1354 (1982) C. Y. Ma, J. L. White, F. C. Weissert, and K. Min, J. Non-Newtonian Fluid Mech., 17, 275 (1985); ibid., Polym. Comp., 6, 215 (1985) J. L. White, Y. Wang, A. I. Isayev, N. Nakajima, F. C. Weissert, and K. Min, Rubber Chem. Technol., 60, 337 (1987) K. C. Shin, J. L. White, R. Brzoskowski, and N. Nakajima, Kautschuk Gummi Kunststoffe, 43, 181 (1990) G. J. Osanaiye, A. I. Leonov, and J. L. White, J. Non-Newtonian Fluid Mech., 49, 87 (1993); ibid, Rubber Chem. Technol., 68, 50 (1995) L. L. Li and J. L. White, Rubber Chem. Technol., 69, 628 (1996) K. J. Kim and J. L. White, J. Non-Newtonian Fluid Mechanics, 66, 257 (1996) T. Araki and J. L. White, Polym. Eng. Sci., 38, 616 (1998); ibid., Nihon Reoroji Gakkaishi, 25, 125 (1997) K. J. Kim and J. L. White, Polym. Eng. Sci., 39, 2189 (1999) K. J. Kim and J. L. White, J. Polym. Sci., Polym. Phys., 37, 1787 (1999) A. Einstein, Ann. Phys., 19, 289 (1906); ibid, 34, 591 (1911); A. Einstein, The Theory of the Brownian Movement, Dover, New York (1956) G. G. Stokes, Trans. Camb. Phil. Soc., 9, 8 (1851) E. Guth, J. Appl. Phys., 16, 20 (1945) E. Guth and R. Simha, Kolloid Z., 74, 266 (1936) G. K. Batchelor, J. Fluid Mech., 41, 545 (1970) J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs (1965) R. Simha, J. Appl. Phys., 23, 1020 (1952) J. Happel, J. Appl. Phys., 28, 1288 (1957) N. A. Frankel and A. Acrivos, Chem. Eng. Sci., 22, 847 (1967) H. Tanaka and J. L. White, J. Non-Newt. Fluid Mech., 7, 333 (1980) M. Mooney, J. Colloid Sci., 6, 162 (1951) G. B. Jeffery, Proc. Roy. Soc. (London), A102, 161 (1922) G. I. Taylor, Proc. Roy. Soc. (London), A103, 58 (1923) G. I. Taylor, Proc. Roy. Soc. (London), A138, 41 (1932); ibid., A146, 501 (1934) B. J. Trevelyan and S. G. Mason, J. Colloid Sci., 6, 354 (1951) S. G. Mason and R. J. Manley, Proc. Roy. Soc. (London), A238, 117 (1956) W. Bartok and S. G. Mason, J. Colloid Sci., 12, 243 (1957) H. L. Goldsmith and S. G. Mason, J. Fluid Mech., 12, 88 (1962) G. K. Batchelor, J. Fluid Mech., 46, 813 (1971) J. D. Goddard, J. Non-Newt. Fluid Mech., 1, 1 (1976) J. G. Oldroyd, Proc. Camb. Phil. Soc., 43, 100, 383, 396, 521 (1947) A. Slibar and P. R. Pasley in Second Order Effects in Elastricity, Plasticity and Fluid Dynamics, M. Reiner and D. Abir (Eds.) MacMillen, New York (1964)

102

49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87.

2 Polymer-Particle Filler Systems

J. L. White, J. Non-Newt. Fluid Mech., 5, 177 (1979); J. L. White and V. M. Lobe, Rheol. Acta, 21, 167 (1982) Y. Suetsugu and J. L. White, J. Non-Newt. Fluid Mech., 14, 121 (1984) S. Montes and J. L. White, J. Non-Newt. Fluid Mech., 49, 277 (1993) A. I. Leonov, J. Rheology, 34, 1039 (1990) J. L. White and C. H. Suh, J. Non-Newt. Fluid Mech., 62 175 (1996) J. L. White, K. J. Kim, and D. N. Robinson, J. Non-Newt. Fluid Mech., 83, 19 (1999) D. N. Robinson, K. J. Kim, and J. L. White, J. Appl. Mech., 69, 641 (2002) H. Eilers, Kolloid Z., 97, 313 (1941) V. Vand, J. Phys. Colloid Chem., 52, 277 (1948) S. G. Ward and R. L. Whitmore, Brit. J. Appl. Phys., 1, 286 (1950) P. Y. Cheng and H. K. Schachman, J. Polym. Sci., 16, 19 (1955) G. Segre and A. Silberberg, J. Colloid Sci., 18, 312 (1963) O. Reynolds, Phil Mag. (5), 20, 469 (1885) H. Freundlich and H. L. Roder, Trans. Faraday Soc., 34, 308 (1938) A. B. Metzner and M. Whitlock, Trans. Soc. Rheology, 2, 239 (1958) R. L. Hoffman, Trans. Soc. Rheology, 16, 155 (1972) F. Nazem and C. T. Hill, Trans. Soc. Rheology, 18, 87 (1972) R. O. Maschmeyer and C. T. Hill, Advances Chem. Series, 134, 95 (1974) J. M. Charrier and J. M. Rieger, Fiber Sci. Technol., 7, 161 (19784) Y. Oyanagi and Y. Yamaguchi, J. Soc. Rheology Japan, 3, 64 (1975) R. O. Maschmeyer and C. T. Hill, Trans. Soc. Rheology, 21, 195 (1977) Y. Chan, J. L. White, and Y. Oyanagi, J. Rheol., 22, 507 (1978) Y. Chan, J. L. White, and Y. Oyanagi, Polym. Eng. Sci., 18, 268 (1978) J. L. White, Y. Chan, and Y. Oyanagi, J. Soc. Rheol Japan, 6, 1 (1978) L. Czarnecki and J. L. White, J. Appl. Polym. Sci., 25, 1217 (1980) J. E. O’Connor, Rubber Chem. Technol., 56, 445 (1977) B. Franzen, C. Klason, J. Kubat, and T. Kitano, Composites, 20, 65 (1989) K. Shon and J. L. White, Polym. Eng. Sci., 39, 1757 (1999) O. L. Forgacs and S. G. Mason, J. Colloid Sci., 14, 457 (1959) I. M. Ward and D. W. Hadley, An Introduction to the Mechanical Properties of Solid Polymers, Wiley, New York (1993) H. Brody and I. M. Ward, Polym. Eng. Sci., 11, 139 (1971) P. H. Hermans and P. Platzek, Kolloid Z., 88, 68 (1939) F. H. Muller, Kolloid Z., 95, 138 (1941) J. J. Hermans, P. H. Hermans, D. Vermaas, and A. Weidinger, Rec. Tran. Chim., 65, 427 (1946) J. L. White and B. A. Knutsson, Polym. Eng. Rev., 2, 71 (1982) J. L. White and J. E. Spruiell, Polym. Eng. Sci., 21, 859 (1981) S. G. Advani and C. L. Tucker, J. Rheol., 31, 751 (1987) R. G. Schierding, J. Composite Mat., 8, 195 (1968) K. Yoshida, G. Budiman, Y. Okayama, and T. Kitao, Seni Gakkaishi, 31, T-355 (1975)

References

88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120.

103

H. Menendez and J. L. White, Polym. Eng. Sci., 24, 1051 (1984) S. Lim, T. Kikutani, J. L. White, and T. Kyu, Adv. Polym. Technol., 8, 325 (1988) S. Lim and J. L. White, J. Rheol., 34, 343 (1990) A. Kelly, Strong Solids, Clarendorn Press, Oxford (1966) L. Mullins, J. Phys. Coll. Chem., 54, 239 (1950) L. Mullins and R. W. Whorlow, Trans IRI, 27, 55 (1950) L. Mullins in The Chemistry and Physics of Rubberlike Substances, L. Bateman (Ed.) MacLaren and Sons, London (1963) J. H. Bachmann, J. W. Sellers, M. P. Wagner, and R. F. Wolf, Rubber Chem. Technol., 32, 1286 (1959) A. R. Payne, R. E. Whittaker, Rubber Chem. Technol., 44, 440 (1971); A. R. Payne, Rubber Plast. Age, 42, 963 (1961) S. Ahn and J. L. White, J. Appl. Polym. Sci. 91, 651 (2004) H. Kim, K. J. Kim, S. M. Kwon, P. P. Kundu, B. C. Jo, B. H. Lee, D. S. Lee, and S. Choe, J. Appl. Polym. Sci., 86, 2041 (2002) S. Kwon, K. J. Kim, H. Kim, P. P. Kundu, T. J. Kim, Y. K. Lee, B. H. Lee, and S. Choe, Polymer, 43, 6901 (2002) Y. Suetsugu, Int. Polym. Process., 5, 184 (1990) K. J. Kim and J. L. White, Composite Interfaces, 9, 541 (2002) K. J. Kim, and J. L. White, J. Ind. Eng. Sci., 6, 262 (2000) P. Rauline (Michelin), US Patent (filed Feb. 20, 1992) 5,227,425 (1993) C. H. Suh and J. L. White, Polym. Eng. Sci., 32, 1521 (1996) J. Menczel and J. Varga, J. Therm. Analys, 28, 161 (1983) A. M. Chatterjee and F. P. Price, J. Polym. Sci.:Polym Phys. Ed., 13, 2385 (1975) J. I. Velasco, J. A. De Saja, and A. B. Martínez, Revista de Plasticos Modernos num., 477, 271 (1996) F. L. Binsbergen, J. Polym. Sci.: Polym. Symp., 59, 11 (1977) M. Fujiyama, Int. Polym. Process., 13, 284 (1998) U. Hofmann, K. Endell, and D. Wilm, Z. Kryst, 86, 340 (1933); Angew. Chem., 47, 539 (1934) C. R. Smith, J. Amer. Chem. Soc., 56, 1561 (1934); US Patent (filed July 5, 1934) 2,033,856 (1936) J. W. Jordan, J. Phys. Coll. Chem., 53, 284 (1949) I. Barshad, Proc. Soil Soc. Amer., 16, 176 (1952) A. Weiss, Angew. Chem. Int. Ed., 2, 134 (1963) M. M. Mortland, S. Shobai, and S. A. Boyd, Clays and Clay Minerals, 34, 581 (1986) A. Blumstein, J. Polym. Sci. A, 3, 2653 (1965) D. H. Soloman and B. C. Loft, J. Appl. Polym. Sci., 12, 1253 (1968) Y. Fukushima, Clays and Clay Minerals, 32, 320 (1984); Y. Fukushima, A. Okada, M. Kawasumi, T. Kurauchi, and O. Kamigato, Clay Minerals, 23, 27 (1988) Y. Kojima, A. Usuki, M. Kawasumi, A. Okada, T. Kurauchi, and O. Kamigaito, J. Polym. Sci.: Part A: Polym. Chemistry, 31, 983 (1993) Y. Kojima, A. Usuki, M. Kawasumi, A. Okada, T. Kurauchi, and O. Kamigaito, J. Appl. Polym. Sci., 49, 1259 (1993)

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121. A. Usuki, Y. Kojima, M. Kawasumi, A. Okada, Y. Fukushima, T. Kurauchi, and O. Kamigaito, J. Mater. Res., 8(5), 1179 (1993) 122. Y. Kojima, A. Usuki, M. Kawasumi, A. Okada, Y. Fukushima, T. Kurauchi, and O. Kamigaito, J. Mater. Res., 8(5), 1185 (1993) 123. Y. Kojima, A. Usuki, M. Kawasumi, A. Okada, T. Kurauchi, and O. Kamigaito, J. Polym. Sci.: Part A: Polym. Chemistry, 31, 1755 (1993) 124. Y. Kojima, T. Matsuoka, H. Takahashi, and T. Kurauchi, J. Appl. Polym. Sci., 51, 683 (1994) 125. Y. Kojima, A. Usuki, M. Kawasumi, A. Okada, T. Kurauchi, O. Kamigaito, and K. Kaji, J. Polym. Sci.: Part B: Polym. Phys., 32, 625 (1994) 126. K. Yano, A. Usuki, A. Okada, T. Kurauchi, and O. Kamigaito, J. Polym. Sci.:Part A: Polym. Chemistry, 31, 2493 (1993) 127. D. C. Lee and L. W. Jang, J. Appl. Polym. Sci., 68, 1997 (1998) 128. P. B. Messersmith and E. P. Giannelis, Chem. Mat., 5(8), 1064 (1993) 129. R. A. Vaia, K. D. Jandt, E. J. Kramer, and E. P. Giannelis, Macromolecules, 28, 8080 (1995) 130. M. B. Ko, M. Park, J. Kim, and C. R. Choe, Korea Polym. J., 8, 95 (2000) 131. D. Wang, D. Parlow, Q. Yao, and C. A. Wilkie, J. Vinyl Additive Technol., 7, 203 (2001) 132. X. Liu and Q. Wu, Macromol. Mater. Eng., 287, 180 (2002) 133. Y. Kim and J. L. White, J. Appl. Polym. Sci., 90, 1581 (2003), ibid, 92, 1061 (2004) 134. R. H. Baughman, A. A. Zakhidov, and W. A. de Heer, Science, 297, 787 (2002) 135. M. S. P. Shaffer and A. H. Windle, Adv. Mater., 11, 937 (1999) 136. X.-Y. Gong, J. Liu, S. Baskaran, R. D. Voise, and J. S. Young, Chem. Mater., 12, 1049 (2000) 137. Z.-X. Jin, K. P. Pramoda, G.-Q. Xu, and S. H. Goh, Chem. Phys. Lett., 337, 43 (2001) 138. S. Kumar, T. D. Dang, F. E. Arnold, A. R. Bhattacharyya, B. G. Min, X.-F. Zhang, R. A. Vaia, C. Park, W. W. Adams, R. H. Hauge, R. E. Smalley, S. Ramesh, and P. A. Willis, Macromolecules, 35, 9039 (2002) 139. T. V. Sreekumar, T. Liu, B. G. Min, H. Guo, S. Kumar, R. H. Hauge, and R. E. Smalley, Adv. Mater, 16, 58 (2004) 140. C. Park and E. J. Siochi, Ninth International Conference on Composite Engineering (ICCE/9), p. 599, San Diego (July 2002)