**2. Comparison of interaction effects in long-chain branched and linear polymeric matrices**

Interparticle interactions take place when the filler concentration increases. This includes an increase of particles per unit volume, which come into contact during flow. Furthermore, the rotation and migration of particles during flow, as well as the formation and breaking of aggregates, produces additional dissipative effects, which lead to an increase in the viscosity [1]. The mathematical consideration of interaction effects, which are dependent on the size, size distribution and morphology of the filler particles, as well as the filler volume concentration and the applied shear rate or shear stress, which is affected by volume output and the flow channel geometry in one general approach that describes the flow behavior of suspensions, is subject of this chapter. In addition to the particle properties and process conditions, the molecular structure of the matrix polymer has an influence on the formation of interparticle interaction effects and thus also on the flow behavior of the suspension. Linear and long-chain branched polymers differ in both shear thinning flow behavior and nonlinear, steady-state viscoelastic melt properties. The elastic melt properties, defined as the ratio of normal stress difference and shear stress, influence the particle migration during flow. The resulting differences in flow behavior are illustrated below using a long-chain branched LDPE and a linear PP as polymer matrices. In order to characterize the rheological properties of the particle-filled polymer melts, the fillers must be uniformly dispersed into the polymer matrix by means of suitable compounding techniques. The fillers mentioned in this chapter (**Figure 6**) can be classified based on their geometrical shape into the following categories:


interaction exponent remains constant at a value of one. The interaction exponent decreases

**Figure 4.** Typical function of the correlation between consistency index of a suspension and volumetric filler concentration.

A general mathematical description of the interaction exponent over the entire range of negligible interparticle interactions with the transition to non-negligible interparticle interactions provides the following equation, which is called the *generalized interaction function* [18]:

> (1 + ( *K*\_\_\_*c <sup>K</sup>*∗) *a* ) \_\_*b a*

(8)

only if interparticle interactions occur (**Figure 5**).

*χ* = \_\_\_\_\_\_\_\_\_\_ <sup>1</sup>

**Figure 5.** Interaction exponent of a suspension as a function of the consistency index.

K\* = transitional consistency index [Pa sn]

a, b = adjustable parameters [−]

where,

140 Polymer Rheology

• Fibrous wood flour (big fraction), wood fibers

The particle size distribution was determined using the Mastersizer 3000 (Malvern Instruments GmbH) via laser diffraction. The preparations of all formulations as shown in the following section were carried out by compounding on a corotating twin screw extruder (Brabender DSE 20) with a mass temperature of 190°C and mass output of 5 kg/h. Temperature profile: feed zone 170°C; plastification zone 180°C; conveying zone 185°C; mixing zone 185°C; extrusion zone 190°C.

#### **2.1. Low-density polyethylene**

The polymer matrix of all following compounds in this section is a low-density polyethylene (LDPE), which is highly branched and exhibits shear thinning flow behavior.

**Figure 6.** Light transmitted microscopy of spherical, plate-like and fibrous particles.

**Figure 7** shows the flow functions (shear stress as a function of shear rate) of unfilled LDPE and LDPE filled with different volumetric concentrations of wood fibers. The volume concentrations correspond to typical mass fractions and have been discussed in previous studies [14].

in shear stress of the suspension to the shear stress of the polymer matrix is the highest for the fibrous particles (*wood fibers*), slightly lower for the plate-like particles (*natural graphite*), and the lowest for the spherical particles (*glass beads*). With increasing shear rates, the impact of the

**Figure 8.** Flow functions of LDPE filled with 46 vol% of different fillers; glass beads (d50 = 346 μm), natural graphite

Interparticle Interaction Effects in Polymer Suspensions http://dx.doi.org/10.5772/intechopen.75207 143

(d50 = 267 μm), wood fibers (d50 = 527 μm) in comparison with the unfilled LDPE (T = 190°C).

In order to describe the impact of the volumetric filler concentration on the consistency index of particle-filled LDPE for various fillers in different size fractions, Eq. (7) has been used. **Figure 9** shows that experimental data (*symbols*) can be excellently fitted (*lines*) on the basis

At low filler concentrations up to approx. 20 vol%, the particle properties, for example, particle size and morphology, have no significant impact on the extent of the consistency index

**Figure 9.** Consistency index as a function of volumetric filler concentration for LDPE filled with various fillers with different particle sizes, (T = 190°C); (a) glass beads (d50 = 346 μm), natural graphite (d50 = 267 μm), wood fibers

(d50 = 527 μm); (b) glass beads (d50 = 60 μm), natural graphite (d50 = 77 μm), wood flour (d50 = 110 μm).

filler morphology on the flow behavior decreases.

of this equation.

All flow functions exhibit shear thinning flow behavior, which can be mathematically described by the power law of Ostwald/de Waele. With increasing volumetric content of the wood fibers, the level of the flow functions is shifted to higher values. Accordingly, the consistency indices (K) increase with the increasing volume fraction of the filler particles. The extent of the consistency increase depends on the size and geometrical shape of the filler particles. **Figure 8** shows flow functions of various filled LDPE systems at a fixed loading of 46 vol% to illustrate the impact of the filler morphology on the flow behavior. At low shear rates, the relative increase

**Figure 7.** Flow functions of LDPE filled with varying volume fractions of wood fibers (d50 = 527 μm).

**Figure 8.** Flow functions of LDPE filled with 46 vol% of different fillers; glass beads (d50 = 346 μm), natural graphite (d50 = 267 μm), wood fibers (d50 = 527 μm) in comparison with the unfilled LDPE (T = 190°C).

in shear stress of the suspension to the shear stress of the polymer matrix is the highest for the fibrous particles (*wood fibers*), slightly lower for the plate-like particles (*natural graphite*), and the lowest for the spherical particles (*glass beads*). With increasing shear rates, the impact of the filler morphology on the flow behavior decreases.

In order to describe the impact of the volumetric filler concentration on the consistency index of particle-filled LDPE for various fillers in different size fractions, Eq. (7) has been used. **Figure 9** shows that experimental data (*symbols*) can be excellently fitted (*lines*) on the basis of this equation.

At low filler concentrations up to approx. 20 vol%, the particle properties, for example, particle size and morphology, have no significant impact on the extent of the consistency index

**Figure 9.** Consistency index as a function of volumetric filler concentration for LDPE filled with various fillers with different particle sizes, (T = 190°C); (a) glass beads (d50 = 346 μm), natural graphite (d50 = 267 μm), wood fibers (d50 = 527 μm); (b) glass beads (d50 = 60 μm), natural graphite (d50 = 77 μm), wood flour (d50 = 110 μm).

**Figure 7.** Flow functions of LDPE filled with varying volume fractions of wood fibers (d50 = 527 μm).

**Figure 7** shows the flow functions (shear stress as a function of shear rate) of unfilled LDPE and LDPE filled with different volumetric concentrations of wood fibers. The volume concentrations correspond to typical mass fractions and have been discussed in previous studies [14]. All flow functions exhibit shear thinning flow behavior, which can be mathematically described by the power law of Ostwald/de Waele. With increasing volumetric content of the wood fibers, the level of the flow functions is shifted to higher values. Accordingly, the consistency indices (K) increase with the increasing volume fraction of the filler particles. The extent of the consistency increase depends on the size and geometrical shape of the filler particles. **Figure 8** shows flow functions of various filled LDPE systems at a fixed loading of 46 vol% to illustrate the impact of the filler morphology on the flow behavior. At low shear rates, the relative increase

**Figure 6.** Light transmitted microscopy of spherical, plate-like and fibrous particles.

142 Polymer Rheology

increase. At higher filler concentrations, the differences between the filler types are substantial. High aspect ratio particles such as wood fibers and natural graphite, regardless of their size, have a significantly greater impact on the consistency index than the spherical glass beads. Generally, the viscosity of a suspension and thus the consistency index exhibit higher values with decreasing particle size at a constant filler volume fraction. An exception can be observed here with the wood fillers. The small particle fraction contains a considerable amount of wood dust, which has a much lower aspect ratio as compared to the larger wood fibers. Since the impact of the aspect ratio on the consistency index increase is stronger in comparison with the particle size, the larger fibrous wood fillers cause a higher increase in consistency than the smaller particles of wood flour.

The *generalized interaction function* (Eq. (8)) is used in order to describe the interaction exponent as a function of the consistency index, taking into account the transition from negligible interparticle interactions to the domain of non-negligible interactions. **Figure 10** shows that experimental data (symbols) can be fitted (lines) on the basis of this equation with high accuracy.

The interaction function illustrates that glass beads in LDPE exhibit less interparticle interactions in comparison with particles with larger aspect ratio. In particular, large glass beads even show a range of negligible interparticle interactions with a transition to non-negligible interactions. Regardless of type and size, high aspect ratio particles have a characteristic relationship between consistency index and interaction exponent, which is distinctive of the polymer matrix LDPE.

Based on the *generalized interaction function* (Eq. (8)) and the relationship between consistency and volumetric filler concentration (Eq. (7)), the shift factor B can be derived as a function of the volume fraction for variable shear stresses (Eq. (6)). **Figure 11** comparatively illustrates the influence of the filler volume concentration and the applied shear stress for various fillers in

different size fractions on the shift factor B. On the basis of the shift factor B, the flow behavior of polymer suspensions can be estimated for arbitrary volume concentrations and shear

**Figure 11.** Shift factor B as a three-dimensional function of filler content and applied shear stress for LDPE filled with various fillers; (a) glass beads (d50 = 60 μm), (b) glass beads (d50 = 346 μm), (c) natural graphite (d50 = 77 μm), (d) natural

The polymer matrix of all following compounds in this section is a polypropylene (PP), which

*<sup>n</sup>*<sup>0</sup> (9)

Interparticle Interaction Effects in Polymer Suspensions http://dx.doi.org/10.5772/intechopen.75207 145

stresses or shear rates on the basis of the following equation:

graphite (d50 = 267 μm), (e) wood flour (d50 = 110 μm), (f) wood fibers (d50 = 527 μm).

exhibit a linear molecular structure and shear thinning flow behavior.

*τ<sup>c</sup>* = *Bn*<sup>0</sup> ⋅ *τ*<sup>0</sup> = *Bn*<sup>0</sup> ⋅ *K*<sup>0</sup> ⋅ *γ*̇

**2.2. Polypropylene**

**Figure 10.** Interaction exponent as a function of consistency index of LDPE filled with various fillers (T = 190°C).

**Figure 11.** Shift factor B as a three-dimensional function of filler content and applied shear stress for LDPE filled with various fillers; (a) glass beads (d50 = 60 μm), (b) glass beads (d50 = 346 μm), (c) natural graphite (d50 = 77 μm), (d) natural graphite (d50 = 267 μm), (e) wood flour (d50 = 110 μm), (f) wood fibers (d50 = 527 μm).

different size fractions on the shift factor B. On the basis of the shift factor B, the flow behavior of polymer suspensions can be estimated for arbitrary volume concentrations and shear stresses or shear rates on the basis of the following equation:

$$
\pi\_{\iota} = B^{u\_{\iota}} \cdot \pi\_{\iota} = B^{u\_{\iota}} \cdot K\_{\iota} \cdot \dot{\gamma}^{u\_{\iota}} \tag{9}
$$

#### **2.2. Polypropylene**

increase. At higher filler concentrations, the differences between the filler types are substantial. High aspect ratio particles such as wood fibers and natural graphite, regardless of their size, have a significantly greater impact on the consistency index than the spherical glass beads. Generally, the viscosity of a suspension and thus the consistency index exhibit higher values with decreasing particle size at a constant filler volume fraction. An exception can be observed here with the wood fillers. The small particle fraction contains a considerable amount of wood dust, which has a much lower aspect ratio as compared to the larger wood fibers. Since the impact of the aspect ratio on the consistency index increase is stronger in comparison with the particle size, the larger fibrous wood fillers cause a higher increase in

The *generalized interaction function* (Eq. (8)) is used in order to describe the interaction exponent as a function of the consistency index, taking into account the transition from negligible interparticle interactions to the domain of non-negligible interactions. **Figure 10** shows that experimental data (symbols) can be fitted (lines) on the basis of this equation with high

The interaction function illustrates that glass beads in LDPE exhibit less interparticle interactions in comparison with particles with larger aspect ratio. In particular, large glass beads even show a range of negligible interparticle interactions with a transition to non-negligible interactions. Regardless of type and size, high aspect ratio particles have a characteristic relationship between consistency index and interaction exponent, which is distinctive of the poly-

Based on the *generalized interaction function* (Eq. (8)) and the relationship between consistency and volumetric filler concentration (Eq. (7)), the shift factor B can be derived as a function of the volume fraction for variable shear stresses (Eq. (6)). **Figure 11** comparatively illustrates the influence of the filler volume concentration and the applied shear stress for various fillers in

**Figure 10.** Interaction exponent as a function of consistency index of LDPE filled with various fillers (T = 190°C).

consistency than the smaller particles of wood flour.

accuracy.

144 Polymer Rheology

mer matrix LDPE.

The polymer matrix of all following compounds in this section is a polypropylene (PP), which exhibit a linear molecular structure and shear thinning flow behavior.

**Figure 12** shows the flow functions (shear stress as a function of shear rate) of PP filled with different volumetric concentrations of wood flour, as well as of the unfilled polymer matrix.

As with particle-filled LDPE, all PP-based flow functions exhibit shear thinning flow behavior. In the observed shear rate range, the flow behavior can also be described very well by the power law. Compared to the LDPE-based compounds, the flow behavior index (n) of the PP-based formulations has slightly smaller values and thus a higher pseudoplasticity.

The influence of the filler morphology on the flow behavior of PP filled with different filler types at a fixed loading of 46 vol% is shown in **Figure 13**. Compared to long-chain branched LDPE, the impact of filler type on the flow function of linear PP is lower. Furthermore, it is obvious that the flow behavior index is only slightly affected by the different aspect ratio of the filler types.

On the basis of (Eq. (7)), the impact of the volumetric filler concentration on the consistency index of particle-filled PP has been described. **Figure 14** shows that experimental data (*symbols*) can be excellently fitted (*lines*).

Over the entire range of volume fraction, the differences between the filler types are substantial regarding the impact on the consistency index increase. Basically, a strong influence can be observed of the aspect ratio of the filler particles on the consistency increase. High aspect ratio particles such as wood fibers and natural graphite, regardless of their size, have a significantly greater impact on the consistency index than the spherical glass beads, especially at high filler contents.

**Figure 15** presents the correlation between interaction exponent and consistency index, which have been mathematically described by the *generalized interaction function* (Eq. (8)). This contains all fillers; wood fibers/flour, natural graphite and glass beads, in the entire concentration range from 6 to 56 vol%, in each case in two particle size fractions.

It is a remarkable fact that there is a characteristic relationship between interaction exponent and consistency index for all PP-based compounds, regardless of filler type, size and volume

fraction. On the basis of this characteristic correlation, interparticle interactions in PP-based suspensions can be universally described with only one set of parameters of the generalized interaction function. The parameters that are generally valid for particle-filled PP to describe

**Figure 14.** Consistency index as a function of volumetric filler concentration for PP filled with various fillers with different particle sizes, (T = 200°C); (a) glass beads (d50 = 346 μm), natural graphite (d50 = 267 μm), wood fibers (d50 = 527 μm); (b)

**Figure 13.** Flow functions of PP filled with 46 vol% of different fillers; glass beads (d50 = 346 μm), natural graphite

Interparticle Interaction Effects in Polymer Suspensions http://dx.doi.org/10.5772/intechopen.75207 147

(d50 = 267 μm), wood fibers (d50 = 527 μm) in comparison with the unfilled LDPE (T = 200°C).

the interaction exponent as a function of consistency (Eq. (8)) index are:

glass beads (d50 = 60 μm), natural graphite (d50 = 77 μm), wood flour (d50 = 110 μm).

K<sup>∗</sup> = 16706 Pa sn

a = 4.076

b = 0.385

**Figure 12.** Flow functions of PP filled with varying volume fractions of wood flour (d50 = 110 μm), (T = 200°C).

**Figure 12** shows the flow functions (shear stress as a function of shear rate) of PP filled with different volumetric concentrations of wood flour, as well as of the unfilled polymer matrix. As with particle-filled LDPE, all PP-based flow functions exhibit shear thinning flow behavior. In the observed shear rate range, the flow behavior can also be described very well by the power law. Compared to the LDPE-based compounds, the flow behavior index (n) of the

The influence of the filler morphology on the flow behavior of PP filled with different filler types at a fixed loading of 46 vol% is shown in **Figure 13**. Compared to long-chain branched LDPE, the impact of filler type on the flow function of linear PP is lower. Furthermore, it is obvious that the flow behavior index is only slightly affected by the different aspect ratio of the filler types. On the basis of (Eq. (7)), the impact of the volumetric filler concentration on the consistency index of particle-filled PP has been described. **Figure 14** shows that experimental data (*sym-*

Over the entire range of volume fraction, the differences between the filler types are substantial regarding the impact on the consistency index increase. Basically, a strong influence can be observed of the aspect ratio of the filler particles on the consistency increase. High aspect ratio particles such as wood fibers and natural graphite, regardless of their size, have a significantly greater impact on the consistency index than the spherical glass beads, especially

**Figure 15** presents the correlation between interaction exponent and consistency index, which have been mathematically described by the *generalized interaction function* (Eq. (8)). This contains all fillers; wood fibers/flour, natural graphite and glass beads, in the entire concentration

It is a remarkable fact that there is a characteristic relationship between interaction exponent and consistency index for all PP-based compounds, regardless of filler type, size and volume

**Figure 12.** Flow functions of PP filled with varying volume fractions of wood flour (d50 = 110 μm), (T = 200°C).

range from 6 to 56 vol%, in each case in two particle size fractions.

PP-based formulations has slightly smaller values and thus a higher pseudoplasticity.

*bols*) can be excellently fitted (*lines*).

at high filler contents.

146 Polymer Rheology

**Figure 13.** Flow functions of PP filled with 46 vol% of different fillers; glass beads (d50 = 346 μm), natural graphite (d50 = 267 μm), wood fibers (d50 = 527 μm) in comparison with the unfilled LDPE (T = 200°C).

**Figure 14.** Consistency index as a function of volumetric filler concentration for PP filled with various fillers with different particle sizes, (T = 200°C); (a) glass beads (d50 = 346 μm), natural graphite (d50 = 267 μm), wood fibers (d50 = 527 μm); (b) glass beads (d50 = 60 μm), natural graphite (d50 = 77 μm), wood flour (d50 = 110 μm).

fraction. On the basis of this characteristic correlation, interparticle interactions in PP-based suspensions can be universally described with only one set of parameters of the generalized interaction function. The parameters that are generally valid for particle-filled PP to describe the interaction exponent as a function of consistency (Eq. (8)) index are:

$$\begin{aligned} \mathbf{K}^\* &= 16706 \text{ Pa s}^n \\\\ \mathbf{a} &= 4.076 \end{aligned}$$

b = 0.385

**Figure 15.** Interaction exponent as a function of consistency index of PP filled with various fillers (T = 200°C).

On the basis of these parameters, the flow curves of particle-filled PP can be estimated, regardless of the filler type, particle size, and volume fraction. **Figure 16** exemplarily illustrates this on different formulations. The experimental data (symbols) have been excellently fitted (lines), using that generally valid parameter set of the interaction function.

**Figure 17** comparatively illustrates the influence of the filler volume concentration and the applied shear stress for various fillers in different size fractions on the shift factor B, which has been derived on the basis of (Eq. (6)). On the basis of the shift factor B, the flow behavior of polymer suspensions can be estimated for arbitrary volume concentrations and shear stresses or shear rates, respectively.

**3. Conclusion**

individual polymer matrix.

In this chapter, new mathematical models describing interparticle interaction effects in longchain branched and linear polymer matrices have been presented. In the context of studies with variable volumetric filler concentrations, the influence of filler type (morphology and

**Figure 17.** Shift factor B as a three-dimensional function of filler content and applied shear stress for PP filled with various fillers; (a) glass beads (d50 = 60 μm), (b) glass beads (d50 = 346 μm), (c) natural graphite (d50 = 77 μm), (d) natural

Interparticle Interaction Effects in Polymer Suspensions http://dx.doi.org/10.5772/intechopen.75207 149

On the basis of the *generalized interaction function*, the correlation between interaction exponent and consistency index of particle-filled polymer melts can be mathematically described with high accuracy to experimental data. This correlation is characteristic and valid for each

aspect ratio) and particle size on interparticle interactions has been compared.

graphite (d50 = 267 μm), (e) wood flour (d50 = 110 μm), (f) wood fibers (d50 = 527 μm).

**Figure 16.** Comparison of experimental data (symbols) to predicted flow curves (lines) for PP filled with various fillers by using just one parameter set of the interaction function (K\* = 16,706 Pa s<sup>n</sup>, a = 4.076, b = 0.385), (T = 200°C).

**Figure 17.** Shift factor B as a three-dimensional function of filler content and applied shear stress for PP filled with various fillers; (a) glass beads (d50 = 60 μm), (b) glass beads (d50 = 346 μm), (c) natural graphite (d50 = 77 μm), (d) natural graphite (d50 = 267 μm), (e) wood flour (d50 = 110 μm), (f) wood fibers (d50 = 527 μm).
