(.! Introduction
`
`!!+
`
`(5.4)
`
`
`
`DD
`
`ρ
`
`u
` +
`= −∇ + ∇
`p
`g
`τ ρ
`
`t
`The full form of Eq. 5.4 is presented in Table VI of the appendix.
`
`⋅
`
`5.1.2 The Generalized Newtonian Fluid
`
`The viscosity of most polymer melts is shear thinning and temperature dependent.
`The shear thinning effect is defined as the reduction in viscosity at high rates of
`deformation. This phenomenon occurs because at high rates of deformation the
`molecules are stretched out, enabling them to slide past each other with more ease,
`hence, lowering the bulk viscosity of the melt. Figure 5.3 clearly shows the shear
`thinning behavior and temperature dependence of the viscosity for a selected num-
`ber of thermoplastics. The figure also illustrates ranges of rate of deformation that
`are typical for various processing techniques. To take into consideration non-New-
`tonian effects, it is common to use a viscosity model that is a function of the strain
`rate and temperature to calculate the stress tensor 2 in Eq. 5.4:
`,T⋅
`⋅
`= (
`)
`τ η γ γ
`(5.5)
`where η is the viscosity and γ⋅ the strain rate or rate of deformation tensor defined
`by
`
`⋅
`γ = ∇ + ∇u
`
`ut
`
`(5.6)
`
`where ∇u represents the velocity gradient tensor. This model describes the Gener-
`alized Newtonian Fluid. In Eq. 5.5, γ⋅ is the magnitude of the strain rate tensor and
`can be written as
`γ = 1
`⋅
`2
`where II is the second invariant of the strain rate tensor defined by
`=∑ ∑ γ γ⋅
`⋅
`
`II
`ji
`i
`j
`ij
`The strain rate tensor components in Eq. 5.8 are defined by
`⋅
`γij
`
`(5.7)
`
`(5.8)
`
`(5.9)
`
`II
`
`
`
`
`
`u x
`∂ ∂
`
`j i
`
`+
`
`ux
`∂ ∂
`
`i j
`
`=
`
`The temperature dependence of the polymer’s viscosity is normally factored out as
`⋅
`⋅
`(
`)
`= (
`)
`(
`)
`
`(5.10)
`T
`f T
`,
`η γ
`η γ
`
`2  As will be shown later, this is only true when the elastic effects are negligible during deformation of the
`polymeric material.
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 131
`
`

`

`!!#
`
`5 Rheology of Polymer Melts
`
`Compression molding
`Calendering
`
`Extrusion
`
`Injection molding
`
`Fiber spinning
`
`220 °C
`ABS
`260 °C
`300 °C
`
`340 °C
`260 °C
`
`PA6
`
`300 °C
`
`PC
`
`104
`
`Pa*s
`
`103
`
`102
`
`Viscosity, (cid:100)
`
`101
`101
`
`103
`Shear rate, (cid:97)
`Figure 5.3 Viscosity curves for a selected number of thermoplastics
`
`102
`
`104
`
`1/s
`
`105
`
`) can be approximated using an
`
`)
`(
`1
`1
`) =
`(
`T
`T
`0
`where E0 is the activation energy, T0 a reference temperature, and R the gas con-
`stant. Using this shi! , the viscosity curves measured at diff erent temperatures can
`be translated to generate a master curve at a specifi c temperature. Figure 5.4 [9]
`presents the viscosity of a low density polyethylene with measured values shi! ed
`to a reference temperature of 150 °C. For the shi! in Fig. 5.4, an activation energy
`E0= 54 kJ/mol was used.
`Several models that are used to represent the strain rate dependence of polymer
`melts are presented later in this chapter.
`
` 
`ER
` 
`
`0
`
`exp
`
`T T
`
`0
`
`η η
`
`0 0
`
`where for small variations in temperature, f T(
`exponential function such as
`(
`)
`(
`) =
`(
`)
`(5.11)
`
`f T
`a T T
`exp
`−
`−
`0
`However, a variation in temperature corresponds to a shi! in the time scale. A shi!
`commonly used for semi-crystalline polymers is the Arrhenius shi! , which is
` written as
`) =
`T (
`a T
`
`(5.12)
`
` 
` 
`
`
`
`−
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 132
`
`

`

`
`
`(.! Introduction
`
`!!$
`
`115 ºC
`130 ºC
`150 ºC
`170 ºC
`190 ºC
`210 ºC
`
`Reduced viscosity, log η/aTP
`
`a-s
`
`-3
`
`s-1
`
`4
`
`-2
`-1
`0
`1
`2
`Reduced shear rate, log aT γO
`Figure 5.4  Reduced viscosity curve for a low density polyethylene at a reference temperature
`of 150 °C
`
`1234567
`
`-4
`
`5.1.3 Normal Stresses in Shear Flow
`
`N
`1
`
`=
`
`−
`τ τ
`xx
`yy
`
`=
`
`= −
`
`= −
`
`
`
`
`
`The tendency of polymer molecules to “curl-up” while they are being stretched in
`shear flow results in normal stresses in the fluid. For example, shear flows exhibit
`a deviatoric stress defined by
`⋅
`)⋅
`= (
`(5.13)
`
`τ η γ γ
`xy
`xy
`−τ τ are referred to
` and N
`Measurable normal stress differences,
`N
`τ τ
`2 =
`1 =
`−
`yy
`zz
`xx
`yy
`as the first and second normal stress differences. The first and second normal stress
`differences are material dependent and are defined by
`⋅
`⋅
`(
`)
`T
`ψ γ γ,
`2
`xy
`1
`⋅
`⋅
`(
`)
`T
`N
`ψ γ γ,
`τ τ
`2
`−
`xy
`yy
`zz
`2
`2
`The material functions ψ1 and ψ2 are called the primary and secondary normal
`stress coefficients, and are also functions of the magnitude of the strain rate tensor
`and temperature. The first and second normal stress differences do not change in
`sign when the direction of the strain rate changes. This is reflected in Eqs. 5.14
`and 5.15. Figure 5.5 [9] presents the first normal stress difference coefficient for
`the low density polyethylene melt shown in Fig. 5.4 at a reference temperature of
`150 °C. The second normal stress difference is difficult to measure and is o!en
`approximated by
`⋅
`⋅
`(
`)≈ −
`(
`)
`
`0 1.
`ψ γ
`γψ
`2
`1
`
`(5.14)
`
`(5.15)
`
`(5.16)
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 133
`
`

`

`1/s
`
`!!%
`
`5 Rheology of Polymer Melts
`
`Reduced first normal stress difference coefficient,
`
`log ((cid:115)1/aT
`2)
`
`Pa*s2
`
`Reduced shear rate,
`log (aT*(cid:97)0)
`Figure 5.5  Reduced first normal stress difference coefficient for a low density polyethylene
`melt at a reference temperature of 150 °C
`
`5.1.4 Deborah Number
`
`A useful parameter o!en used to estimate the elastic effects during flow is the
`Deborah number 3, De. The Deborah number is defined by
`= λ
`
`tp
`
`(5.17)
`
`De
`
`3  From the Song of Deborah, Judges 5 : 5 – “The mountains flowed before the Lord.” M. Reiner is credited for
`naming the Deborah number; Physics Today, (January 1964).
`
` 
`
` © Wolfgang Cohnen, 1998. Coyote
`Buttes North 1 Second Wave, Arizona.
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 134
`
`

`

`
`
`(.! Introduction
`
`!!&
`
`where λ is the relaxation time of the polymer and tp is a characteristic process
`time. The characteristic process time can be defi ned by the ratio of characteristic
`die dimension and average speed through the die. A Deborah number of zero
`re presents a viscous fl uid and a Deborah number of ∞ an elastic solid. As the
`Deborah number becomes > 1, the polymer does not have enough time to relax
`during the process, resulting in possible deviations in extrudate dimension or
`irregularities, such as extrudate swell 4, shark skin, or even melt fracture.
`Although many factors aff ect the amount of extrudate swell, fl uid “memory” and
`normal stress eff ects are the most signifi cant ones. However, abrupt changes in
`boundary conditions, such as the separation point of the extrudate from the die,
`also play a role in the swelling or cross section reduction of the extrudate. In prac-
`tice, the fl uid memory contribution to die swell can be mitigated by lengthening
`the land length of the die. This is schematically depicted in Fig. 5.6. A long die land
`separates the polymer from the manifold long enough to allow it to “forget” its past
`shape.
`
`D0
`
`D1
`
`D2
`
`D0
`
`L1
`Figure 5.6 Schematic diagram of extrudate swell during extrusion
`
`L2
`
`Waves in the extrudate may also appear as a result of high speeds during extrusion
`that do not allow the polymer to relax. This phenomenon is generally referred to as
`shark skin and is shown for a high density polyethylene in Fig. 5.7a [10]. It is pos-
`sible to extrude at such high speeds that an intermittent separation of melt and
`inner die walls occurs, as shown in Fig. 5.7b. This phenomenon is o! en referred to
`as the stick-slip eff ect or spurt fl ow and is attributed to high shear stresses between
`the polymer and the die wall. This phenomenon occurs when the shear stress is
`near the critical value of 0.1 MPa [11–13]. If the speed is further increased, a heli-
`cal geometry is extruded, as shown for a polypropylene extrudate in Fig. 5.7c.
`Eventually, the speeds become so high that a chaotic pattern develops, such as the
`one shown in Fig. 5.7d. This well-known phenomenon is called melt fracture. The
`
`4  It should be pointed out that Newtonian fl uids, which do not experience elastic or normal stress eff ects,
`also show some extrudate swell or reduction. A Newtonian fl uid that is being extruded at high shear rates
`reduces its cross-section to 87 % of the diameter of the die, whereas if extruded at very low shear rates, it
`swells to 113 % of the diameter of the die. This swell is due to inertia eff ects caused by the change from the
`parabolic velocity distribution inside the die to the fl at velocity distribution of the extrudate.
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 135
`
`

`

`!!’
`
`5 Rheology of Polymer Melts
`
`(a)
`
`
`
`
`
`
`(b)
`
`
`
`
`
`(c)
`
`
`
`
`
`(d)
`
`Figure 5.7 Various shapes of extrudates under melt fracture
`
`shark skin effect is rarely experienced with linear polymers, which tend to experi-
`ence spurt flow
`It has been reported that the critical shear stress is independent of the melt tem-
`perature, but inversely proportional to the weight average molecular weight [14,
`12]. However, Vinogradov et al. [15] presented results showing that the critical
`stress was independent of molecular weight, except for low molecular weights.
`Dealy and co-workers [16], and Denn [17] provide an extensive overview of various
`melt fracture phenomena that is recommended reading.
`To summarize, the Deborah number and the size of the deformation imposed upon
`the material during processing determine how to most accurately model the sys-
`tem. Figure 5.8 [1] helps visualize the relation between time scale, deformation,
`and applicable model. At small Deborah numbers, the polymer can be modeled as
`a Newtonian fluid, and at very high Deborah numbers the material can be modeled
`as a Hookean solid. In between, the viscoelastic region is divided in two areas: the
`linear viscoelastic region for small deformations, and the non-linear viscoelastic
`region for large deformations. Linear viscoelasticity was briefly discussed in Chap-
`ter 2.
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 136
`
`

`

`
`
`(.$ Viscous Flow Models
`
`!!(
`
`Elasticity
`
`Non-linear viscoelasticity
`
`Newtonian fluid
`
`Deformation
`
`Linear viscoelasticity
`
`Deborah number
`Figure 5.8  Schematic of New tonian, elastic, linear, and non-linear viscoelastic regimes as a
`function of deformation and Deborah number during deformation of polymeric
`materials
`
` (cid:132) 5.2 Viscous Flow Models
`
`Strictly speaking, the viscosity η, measured with shear deformation viscometers,
`should not be used to represent the elongational terms located on the diagonal of
`the stress and strain rate tensors. Elongational fl ows are briefl y discussed later in
`this chapter. A rheologist’s task is to fi nd the models that best fi t the data for the
`viscosity represented in Eq. 5.5. Some of the models used by polymer processors
`on a day-to-day basis to represent the viscosity of industrial polymers are pre-
`sented in the next section.
`
`5.2.1 The Power Law Model
`
`The power law model proposed by Ostwald [18] and de Waale [19] is a simple
`model that accurately represents the shear thinning region in the viscosity versus
`strain rate curve but neglects the Newtonian plateau present at small strain rates.
`The power law model can be written as follows:
`⋅
`= (
`)
`
`m T
`n 1
`−
`γ
`
`(5.18)
`
`η
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 137
`
`

`

`!*)
`
`5 Rheology of Polymer Melts
`
`where m is referred to as the consistency index and n the power law index. The con-
`sistency index may include the temperature dependence of the viscosity such as
`represented in Eq. 5.11, and the power law index represents the shear thinning
`behavior of the polymer melt. Figure 5.9 presents normalized velocity distribu-
`tions inside a tube for a fl uid calculated using the power law model with various
`power law indices. It should be noted that the limits of this model are
`→0 as ⋅
`⋅
` and η ∞ γ
`γ ∞
`→as
`0
`→
`The infi nite viscosity at zero strain rates leads to an erroneous result for regions of
`zero shear rate, such as at the center of a tube. This results in a predicted velocity
`distribution that is fl atter at the center than the experimental profi le. In computer
`simulation of polymer fl ows, this problem is o! en overcome by using a truncated
`model such as
`)
`(
`m T
`0
`)
`(
`)
`(
`⋅
`⋅
`⋅
`⋅
`
`
`(5.19b)
`
`m T0 m T0
`for
`for
`η
`η
`≤
`≤
`=
`=
`
`γ γγ γ
`0
`0
`⋅ ) viscosity. Table 5.1 presents a list of
`where m0 represents a zero shear rate (γ0
`typical power law and consistency indices for common thermoplastics.
`
`for
`
`⋅
`⋅  and
`>
`γ γ
`0
`
`(5.19a)
`
`η
`
`→
`
`η
`
`=
`
`⋅
`n
`γ
`
`1
`−
`
`n = 1.0
`
`n = 0.5
`n = 0.25
`n = 0.1
`n = 0
`
`2.5
`
`2
`
`1.5
`
`1
`
`0.5
`
`vz(r) / v
`
`0
`
`0
`
`0.25
`
`0.75
`
`1
`
`0.5
`r / R
`
`Figure 5.9 Pressure fl ow velo-
`city distributions inside a tube
`for fl uids with various power law
`indices
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 138
`
`

`

`
`
`(.$ Viscous Flow Models
`
`!*!
`
`Table 5.1 Power Law and Consistency Indices for Common Thermoplastics
`m (Pa-sn)
`Polymer
`T (°C)
`Polystyrene
`170
`2.80 × 104
`High density polyethylene
`180
`2.00 × 104
`Low density polyethylene
`160
`6.00 × 103
`Polypropylene
`200
`7.50 × 103
`Polyamide 66
`290
`6.00 × 102
`Polycarbonate
`300
`6.00 × 102
`Polyvinyl chloride
`180
`1.70 × 104
`
`n
`0.28
`0.41
`0.39
`0.38
`0.66
`0.98
`0.26
`
`5.2.2 The Bird-Carreau-Yasuda Model
`
`A model that fits the whole range of strain rates was developed by Bird and Car-
`reau [20] and Yasuda [21]; it contains five parameters:
`
`(5.20)
`
`−(
`)
`n
`1
`
`a/
`
`
`
`
`
`a
`
`⋅
`λγ
`
`=
`
`+
`1
`
`
`η η
`η η
`0
`
`− −
`
`∞ ∞
`
`where η0 is the zero shear rate viscosity, η∞ is an infinite shear rate viscosity, λ is
`a time constant, n is the power law index and a is a constant, which in the original
`Bird-Carreau model is a = 2. In many cases, the infinite shear rate viscosity is neg-
`ligible, reducing Eq. 5.20 to a three parameter model. Equation 5.20 was modified
`by Menges, Wortberg, and Michaeli [22] to include a temperature dependence
`using a WLF relation. The modified model, which is used in commercial polymer
`data banks, is written as follows:
`k a
`T
`⋅
`k a
`γ
`T
`
`1 2
`
`η
`
`=
`
`k
`3
`
`
`
`(5.21)
`
`1
`+
`
`where the shi! aT applies well for amorphous thermoplastics and is written as
`(
`)
`−(
`)
`k
`k
`T k
`.
`.
`8 86
`8 86
`−
`5
`4
`5
`T k
`k
`k
`.
`.
`101 6
`101 6
`+
`−
`+ −
`5
`5
`4
`Table 5.2 presents constants ki for Carreau-WLF (amorphous) and Carreau-Arrhe-
`nius models (semi-crystalline) for various common thermoplastics. In addition to
`the temperature shi!, Menges, Wortberg and Michaeli [22] measured a pressure
`dependence of the viscosity and proposed the following model, which includes
`both temperature and pressure viscosity shi!s:
`(
`)
`(
`)
`T
`T
`8 86
`.
`T T
`p
`.0 02
`.8 86
`
`
`− +
`s
`s
`(
`T T
`p
`T
`..
`6
`.0 02
`
`101
`+
`− +
`101 6
`.
`+
`−
`s
`s
`
`lna
`T =
`
`−
`
`
`
`(5.22)
`
`(
`η
`log
`
`T p
`,
`
`) =
`
`η
`log
`0
`
`+
`
`−
`*
`
`*
`T
`
`−
`
`
`
`)
`
`(5.23)
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 139
`
`

`

`!**
`
`5 Rheology of Polymer Melts
`
`where p is in bar, and the constant 0.02 represents a 0.02 K shi! per bar5. In the
`above equation, the first term represents a shi! between the measured tempe-
`rature T* and the reference temperature Ts. The second term represents the tem-
`perature and pressure shi!s between the actual temperature and the reference
`temperature, as well as between 1 bar and the actual pressure. Hence, in the above
`model a rise in pressure is equivalent to a drop in temperature.
`
`Table 5.2  Constants for Carreau-WLF (Amorphous) and Carreau-Arrhenius (Semi-Crystalline)
`Models for Various Common Thermoplastic
`
`Polymer
`
`k1
`(Pa-s)
` 1777
`Polystyrene
`High density polyethylene 24198
`Low density polyethylene
`  317
`Polypropylene
` 1386
`Polyamide 66
`   44
`Polycarbonate
`  305
`Polyvinyl chloride
` 1786
`
`k2
`(s)
`0.064
`1.38
`0.015
`0.091
`0.00059
`0.00046
`0.054
`
`k3
`
`0.73
`0.60
`0.61
`0.68
`0.40
`0.48
`0.73
`
`k4
`(°C)
`200
`–
`–
`–
`–
`320
`185
`
`k5
`(°C)
`123
`–
`–
`–
`–
`153
` 88
`
`T0
`(°C)
`–
`200
`189
`220
`300
`–
`–
`
`E0
`(J/mol)
`–
` 22272
` 43694
`427198
`123058
`–
`–
`
`5.2.3 The Bingham Fluid
`
`The Bingham fluid is an empirical model that represents the rheological behavior
`of materials that exhibit a “no flow” region below certain yield stresses,τY, such as
`polymer emulsions and slurries. Since these materials flow like a Newtonian liquid
`above the yield stress, the Bingham model can be represented by
`⋅
` or γ
`
`0
`
`ττ= ≤
`η ∞=
`Y
`
`(5.24a)
`
`(5.24b)
`
`≥
`τ τ
`Y
`
`
`
`τ γ
`
`y
`⋅
`
`=
`η µ
`0
`
`+
`
`Here, τ is the magnitude of the deviatoric stress tensor and is computed in the
`same way as in Eq. 5.7.
`
`5.2.4 Elongational Viscosity
`
`In polymer processes, such as fiber spinning, blow molding, thermoforming, foam-
`ing, certain extrusion die flows, and compression molding with specific processing
`conditions, the major mode of deformation is elongational.
`
`5 This constant can be in the range of 0.01 to 0.03.
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 140
`
`

`

`
`
`(.$ Viscous Flow Models
`
`!*+
`
`To illustrate elongational fl ows, consider the fi ber spinning process shown in Fig.
`5.10.
`
`Figure 5.10 Schematic diagram of a fi ber spinning process
`
`F
`
`A simple elongational fl ow is developed as the fi lament is stretched with the follow-
`ing components for the rate of deformation:
`11 = − ⋅⋅
`
`ε
`γ
`⋅
`22 = − ⋅
`ε
`γ
`
`(5.25a)
`
`(5.25b)
`
`
`
`
`
`⋅
`⋅
`2= −
`γ
`ε
`
`(5.25c)
`33
`where ε⋅ is the elongation rate, and the off -diagonal terms of γij
`⋅ are all zero. The
`diagonal terms of the total stress tensor can be written as
`11 = − − ⋅
`p ηε
`σ
`
`(5.26a)
`
`and
`
`22 = − − ⋅
`p ηε
`σ
`
`⋅
`p ηε
`2
`= − +
`
`σ
`33
`
`(5.26b)
`
`(5.26c)
`
`Since the only outside forces acting on the fi ber are in the axial or 3 direction, for
`the Newtonian case, σ11 and σ12 must be zero. Hence,
`p = − εη⋅  and
`
`(5.27)
`
`=⋅
`⋅
`3=
`σ
`
`ε ηη ε
`
`33
`which is known as elongational viscosity or Trouton viscosity [23]. This is analogous
`to elasticity where the following relation between elastic modulus, E, and shear
`modulus, G, can be written
`+(
`)
`2 1 ν
`
`(5.28)
`
`(5.29)
`
`=
`
`EG
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 141
`
`

`

`!*#
`
`5 Rheology of Polymer Melts
`
`where v is Poisson’s ratio. For the incompressibility case, where v = 0.5, Eq. 5.29
`reduces to
`
`EG
`
`= 3
`(5.30)
`Figure 5.11 [24] shows shear and elongational viscosities for two types of poly-
`styrene. In the region of the Newtonian plateau, the limit of 3, shown in Eq. 5.28, is
`quite clear. Figure 5.12 presents plots of elongational viscosities as a function of
`stress for various thermoplastics at common processing conditions. It should be
`emphasized that measuring elongational or extensional viscosity is an extremely
`diffi cult task. For example, in order to maintain a constant strain rate, the spe-
`cimen must be deformed uniformly exponentially. In addition, a molten polymer
`
`µo = 1.7*108 Pa-s
`µo = 1.6*108 Pa-s
`
`Elongational test
`
`Pa-s
`
`(cid:100)o = 5*107
`T= 140 °C
`Polystyrene I
`Polystyrene II
`
`5*102
`
`103
`
`104
`Stress, (cid:111)(cid:11)(cid:3)(cid:109)
`
`(cid:100)o = 5.5*107 Pa-s
`Shear test
`
`105
`
`Pa
`
`5*105
`
`Figure 5.11 Shear and elon-
`gational viscosity curves for two
`types of polystyrene
`
`5*108
`Pa*s
`108
`5*107
`
`107
`5*106
`
`Viscosity
`
`106
`102
`
`6
`
`Pa*s
`
`LDPE
`
`Ethylene-propylene copolymer
`
`PMMA
`
`POM
`
`PA66
`
`Figure 5.12 Elongational
`viscosity curves as a function
`of tensile stress for several
`thermoplastics
`
`Pa
`
`6
`
`4T
`
`5
`ensile stress, log (cid:109)
`
`5
`
`4
`
`3
`
`2
`
`3
`
`Viscosity, log (cid:43)
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 142
`
`

`

`
`
`(.$ Viscous Flow Models
`
`!*$
`
`must be tested completely submerged in a heated, neutrally buoyant liquid at con-
`stant temperature.
`
`5.2.5 Rheology of Curing Thermosets
`
`The conversion or cure dependent viscosity of a curing thermoset polymer
`increases as the molecular weight of the reacting polymer increases. For the vinyl
`ester, whose curing history6 is shown in Fig. 5.13 [25], the viscosity behaves as
`shown in Fig. 5.14 [25]. Hence, a complete model for viscosity of a reacting poly-
`mer must contain the eff ects of strain rate, γ⋅, temperature, T, and degree of cure, c,
`such as
`⋅
`η η γ= (
`,
`
`,T c
`
`)
`
`
`(5.31)
`
`60 °C
`
`50 °C
`
`40 °C
`
`100
`
`Cure time
`
`101
`
`min
`
`102
`
` 
`
` Figure 5.13 Degree of cure as a function
`of time for a vinyl ester at various iso-
`thermal cure conditions
`
`0.8
`
`0.7
`
`0.6
`
`0.5
`
`0.4
`0.3
`
`0.2
`
`0.1
`
`0.0
`10-1
`
`Degree of cure, c
`
`There are no generalized models that include all these variables for thermosetting
`polymers. However, extensive work has been done on the viscosity of polyure-
`thanes [26, 27] used in reaction injection molding processes. An empirical rela-
`tion, sometimes referred to as the Castro-Macosko model, which models the
`viscosity of these mixing-activated polymers, given as a function of temperature
`and degree of cure, is written as
`C C c
`+
`c
`1
`2
`g
`−
`
`
`
`(5.32)
`
` 
`
`c
`
`c
`
`g
`
` 
`
`η η=
`0
`
`e
`
`E
`
`RT
`
`6  A more in-depth view of curing and solidifi cation processes of thermosetting polymers is given in
`Chapter 7.
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 143
`
`

`

`!*%
`
`5 Rheology of Polymer Melts
`
`where E is the activation energy of the polymer, R is the ideal gas constant, T is the
`temperature, cg is the gel point7, c the degree of cure, and C1 and C2 are constants
`that fi t the experimental data. Figure 5.15 shows the viscosity as a function of time
`and temperature for a 47 % MDI-BDO P(PO-EO) polyurethane, and Fig. 5.16 shows
`the viscosity as a function of degree of cure.
`
`60 °C
`
`50 °C
`
`40 °C
`
`101
`
`Pa*s
`
`100
`
`Viscosity, (cid:100)
`
`10-1
`
`0
`
`0.1
`
`0.4
`0.3
`0.2
`0.5
`Degree of cure, c
`
`0.6
`
`0.7
`
`Figure 5.14 Viscosity as a function of
`degree of cure for a vinyl ester at various
`isothermal cure conditions
`
`100
`
`Pa*s
`
`10
`
`(cid:100)
`
`1
`
`0.1
`
`0.01
`0.1
`
`90 °C
`50 °C
`30 °C
`
`Castro-Macosko-model
`fittings
`
`1
`
`Time
`
`10 min
`
`100
`
`Figure 5.15 Viscosity as a function of
`time for a 47 % MDI-BDO P(PO-EO)
`polyurethane at various isothermal cure
`conditions
`
`7 At the gel point the change of the molecular weight with respect to the degree of cure goes to infi nity.
`Hence, it can be said that at this point all the molecules are interconnected.
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 144
`
`

`

`
`
`1000
`600
`
`90 °C
`50 °C
`30 °C
`
`(.$ Viscous Flow Models
`
`!*&
`
`Castro-Macosko-
`model
`
`0.6
`0.2
`Degree of cure, c
`
`1.0
`
` 
`
` Figure 5.16 Viscosity as a function of degree of cure
`for a 47 % MDI-BDO P(PO-EO) polyurethane at various
`isothermal cure conditions
`
`200
`100
`60
`
`20
`10
`
`126
`
`(cid:100)/((cid:100)0*eE/RT)
`
`5.2.6 Suspension Rheology
`
`Particles suspended in a material, such as in fi lled or reinforced polymers, have a
`direct eff ect on the properties of the fi nal article and on the viscosity during pro-
`cessing. Numerous models have been proposed to estimate the viscosity of fi lled
`liquids [28–32]. Most models proposed are a power series of the form8
`
`1= +
`
`a
`a
`2
`φ φ
`+
`1
`2
`
`+
`
`a
`3
`φ
`3
`
`+ …
`
`η η
`
`f
`
`+
`
`τ γ
`
`η
`f =
`
`0
`The linear term in Eq. 5.33 represents the narrowing of the fl ow passage caused by
`the fi ller that is passively entrained by the fl uid and sustains no deformation, as
`shown in Fig. 5.17. For instance, Einstein’s model, which only includes the linear
`term with a1= 2.5, was derived based on a viscous dissipation balance. The quad-
`ratic term in the equation represents the fi rst-order eff ects of interaction between
`the fi ller particles. Geisbüsch suggested a model with a yield stress, where the
`strain rate of the melt increases by a factor κ as
`⋅
`(
`)
`κη κγ
`0
`⋅
`0
`For high deformation stresses, which are typical in polymer processing, the yield
`stress in the fi lled polymer melt can be neglected. Figure 5.18 compares Geis-
`büsch’s experimental data to Eq. 5.33 using the coeffi cients derived by Guth [30].
`
`(5.33)
`
`(5.34)
`
`η ηf 0
`
` = 1 + 2.5ϕ + 14.1ϕ2
`8  The model developed by Guth in 1938 best fi ts experimental:
`However, a full analysis of the fi rst-order particle interactions gives an analytical value for the quadratic
`term of 6.96.
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 145
`
`

`

`!*’
`
`5 Rheology of Polymer Melts
`
`The data and Guth’s model seem to agree well. A comprehensive survey on parti-
`culate suspensions was given by Gupta [33], and one on short-fi ber suspensions by
`Milliken and Powell [34].
`
`z
`
`v(z)
`
`(cid:97)
`
`Particles
`
`v
`
`v
`
`vf(z)
`k(cid:97)
`
`Figure 5.17 Schematic diagram of strain rate increase in a fi lled system
`
`Experimental data
`a1 = 2.5, a2 = 14.1 (Guth, 1938)
`
`1234567
`
`(cid:100)f/(cid:100)0
`
`0
`
`10
`
`%
`
`50
`
`20
`30
`Volume fraction of filler
`Figure 5.18 Viscosity increase as a function of volume fraction of fi ller for polystyrene and
`low density polyethylene containing spherical glass particles with diameters ranging between
`36 μm and 99.8 μm
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 146
`
`

`

`
`
`(.% Simplified Flow Models Common in Polymer Processing
`
`!*(
`
` (cid:132) 5.3  Simplified Flow Models Common
`in Polymer Processing
`
`vk
`
`∆
`
`Many polymer processing operations can be modeled using simplified geometries
`and material models. This section presents several isothermal flow models in sim-
`ple geometries using a Newtonian viscosity and a power-law viscosity as described
`in Eq. 5.18. Although it is very common to simplify analyses of polymer processes
`by assuming isothermal conditions, most operations are non-isothermal because
`they include melting and are o!en influenced by viscous dissipation. Hence, the
`temperature of the polymer melt ranges between Tg (for amorphous polymers) or Tm
`(for semi-crystalline polymers) and the heater temperature Tw (the subscript w is
`o!en used for “wall”) and o!en exceeds Tw due to viscous dissipation. An estimate
`of the maximum temperature rise due to viscous heating is given by
`2
`η 0
`T
`max =
`(5.35)
`
`8
`where v0 represents a characteristic velocity in the flow system, such as plate velo-
`city in a simple shear flow, and k represents the thermal conductivity. To estimate
`the importance of viscous dissipation, the Brinkman number, Br, is o!en computed
`v
`2
`η 0
`)
`(
`k T Tw
`
`−
`or m
`g
`The Brinkman number is the ratio of the heat generated via viscous dissipation
`and the heat conducted out of the system via conduction. A Brinkman number > 1
`implies that the temperature field will be affected by viscous dissipation. The
`choice of temperatures in the denominator of the equation depends on the type of
`material and problem being analyzed.
`
`Br
`
`=
`
`
`
`(5.36)
`
`5.3.1 Simple Shear Flow
`
`v
`
`Simple shear flows, as represented in Fig. 5.19, are very common in polymer pro-
`cessing, such as inside extruders as well as in certain coating flows. The flow field
`in simple shear is the same for all fluids and is described by
`v y
`z = 0
`
`h
`for the velocity distribution and
`Q v hW
`
`= 0
`2
`for the volumetric throughput, where W represents the width of the plates.
`
`(5.37)
`
`(5.38)
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 147
`
`

`

`!+)
`
`5 Rheology of Polymer Melts
`
`y
`
`z
`
`h
`
`u0
`
`5.3.2 Pressure Flow Through a Slit
`
`Figure 5.19 Schematic diagram of a
`simple shear fl ow
`
`The pressure fl ow through a slit, as depicted in Fig. 5.20, is encountered in fl ows
`through extrusion dies or inside injection molds. The fl ow fi eld for a Newtonian
`fl uid is described by
`h
`p
`2
`z (
`) =
`L
`8
`
` 
`
`
`
`2
`
` 
`yh
`
`2
`
`− 
`
`
` 
`
`1
`
` 
`
`µ∆
`
` 
`
`v
`
`y
`
`and
`
`(5.39)
`
`p
`Q Wh
`3
`∆
`=
`L
`12µ
`for the net volumetric throughput. When using the power-law model, the fl ow fi eld
`is described by
`(
`) =
`
`(5.40)
`
`(5.41)
`
`
`
`
`
` 
`
`s
`
`1
`+
`
` 
`yh
`
`2
`
`− 
`
`
` 
`
`1
`
`s
`
` 
`
`h p
`∆
`mL
`2
`
` 
`
` 
`
`h
`+(
`s
`
`2
`
`)
`1
`
` 
`
`v
`
`z
`
`y
`
`and
`
`=1 / , and n the power-law index.
`n
`
`(5.42)
`
`s
`
`
`
` 
`
` 
`
`h p
`Wh
`2
`∆
`)
`+(
`mL
`s
`2
`2
`2
`for the throughput, where s
`
`Q
`
`=
`
`L
`
`h
`
`y
`
`p1
`
`z
`
`(cid:54)p = p1 - p2
`p2
`
` 
`
` Figure 5.20 Schematic diagram of
`pressure fl ow through a slit
`
`5.3.3 Pressure Flow through a Tube – Hagen-Poiseuille Flow
`
`Pressure fl ow through a tube (Fig. 5.21), or Hagen-Poiseuille fl ow, is the type that
`exists inside the runner system in injection molds, as well as in extrusion dies and
`inside the capillary viscometer. For a Newtonian fl uid, the fl ow fi eld inside a tube
`is described by
`R
`2
`z ( ) =
`r
`4
`
` 
`
`
`
`2
`
` 
`rR
`
`− 
`
`
` 
`
`1
`
`p
`L
`
`µ∆
`
`v
`
`(5.43)
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 148
`
`

`

`(.% Simplifi ed Flow Models Common in Polymer Processing
`
`!+!
`
`
`
`and
`
`p
`
`µ4
`
`Q
`
`v
`
`z
`
` 
`
`1
`
` 
`
` 
`
`s
`
`1
`
`s
`
`
`
` 
`
`and
`
` 
`
`Q
`
`=
`
`R p
`R
`π 3
`∆
`mL
`s
`2
`3
`+
`for the throughput.
`
` 
` 
`
`R
`∆
`= π
`
`L
`8
`for the throughput.
`Using the power law model, the fl ow through a tube is described by
`s
`s
`1
`+
`R
`R p
`∆
`− 
`( ) =
`r
`
`mL
`2
`+
`
`(5.44)
`
`(5.45)
`
`(5.46)
`
` 
`
`
`
` 
`rR
`
`L
`
`R
`
`(cid:54)p = p1 - p2
`
`p2
`p1
`Figure 5.21 Schematic diagram of pressure fl ow through a tube
`
`5.3.4 Couette Flow
`
`The Couette device, as depicted in Fig. 5.22, is encountered in bearings and in cer-
`tain types of rheometers. It is also used as a simplifi ed fl ow model for mixers and
`extruders. The Newtonian fl ow fi eld in a Couette device is described by
`R
`r
`2
`−
`( ) =
`r
`r
`1
`κ
`−
`. Using the power law model, the fl ow fi eld inside a Couette device is
`where = R
`Ri
`0 /
`described by
`( ) =
`r
`
` 
`
`
`
`02
`
` 
`
`Ω 2
`
`R
`
`02
`
` 
`
`Ω
`s
`2
`κ
`−
`
`1
`
` 
`
`
`
`2
`
`s
`
`r
`−
`s
`2 1
`−
`
`s
`r
`
`v
`θ
`
`v
`θ
`
`(5.47)
`
`(5.48)
`
`(cid:49)
`
`R0
`Ri
`
` Figure 5.22 Schematic diagram of a Couette device
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 149
`
`

`

`!+*
`
`5 Rheology of Polymer Melts
`
` (cid:132) 5.4 Viscoelastic Flow Models
`
`Viscoelasticity has already been introduced in Chapter 3, based on linear visco-
`elasticity. However, in polymer processing large deformations are imposed on the
`material, requiring the use of non-linear viscoelastic models. There are two types
`of general, non-linear viscoelastic flow models: the differential type and the inte-
`gral type.
`
`5.4.1 Differential Viscoelastic Models
`
`(5.49)
`
` 
`
` 
`
`Differential models have traditionally been the choice for describing the viscoelas-
`tic behavior of polymers when simulating complex flow systems. Many differential
`viscoelastic models can be described using the general form
`(
`) =
`⋅
`⋅
`⋅⋅
`Yτ λτ λ γ τ τ γ λ τ τ η γ
`
`⋅
`⋅
`+
`+
`+
`+
`( )
`1
`2
`3
`0
`1
`where τ 1( ) is the first contravariant convected time derivative of the deviatoric stress
`tensor and represents rates of change with respect to a convected coordinate sys-
`tem that moves and deforms with the fluid. The convected derivative of the devia-
`toric stress tensor is defined as
`τ
`)
`(
`τ
`1( ) =
`− ∇
`(5.50)
`
`t
`The constants in Eq. 5.49 are defined in Table 5.3 for various viscoelastic models
`commonly used to simulate polymer flows. A review by Bird and Wiest [6] provides
`a more complete list of existing viscoelastic models, and Giacomin et al. renewed
`the interest in co-rotational models [35].
`The upper convective model and the White-Metzner model are very similar, with the
`exception that the White-Metzner model incorporates the strain rate effects of
`the relaxation time and the viscosity. Both models provide a first order approxi-
`mation to flows in which shear rate dependence and memory effects are important.
`However, both models predict zero second normal stress coefficients. The Giesekus
`model is molecular-based, non-linear in nature and describes the power law region
`for viscosity and both normal stress coefficients. The Phan-Thien Tanner models
`are based on network theory and give non-linear stresses. Both the Giesekus and
`Phan-Thien Tanner models have been successfully used to model complex flows.
`
`v
`
`t ⋅
`
`τ τ
`+
`
`⋅
`
`∇
`
`v
`
`D D
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 150
`
`

`

`
`
`(.& Viscoelastic Flow Models
`
`!++
`
`Table 5.3 Definition of Constants in Eq. 5.49
`Y
`Constitutive model
`Generalized Newtonian
`1
`Upper convected Maxwell 1
`White-Metzner
`1
`Phan-Thien Tanner-1
`
`(cid:79)1
`0
`(cid:79)1
`λ γ1( )⋅
`
`(cid:79)1
`
`e− ( / )tr
`0 τ
`λ ηε
`
`(cid:79)3
`0
`0
`0
`
`0
`
`0
`
`/ )
`−(
`αλ η1 0
`
`
`(cid:79)2
`0
`0
`0
`
`ξλ
`
`ξλ
`
`12
`
`12
`
`0
`
`Phan-Thien Tanner-2
`
`Giesekus
`
`1
`
`− ( / )λ η τtr
`ε
`0
`
`1
`
`(cid:79)
`
`(cid:79)1
`
`An overview of viscoelastic flow models with a literature review on the subject is
`given by Giacomin et al. [6] and Phan-Thien [36], and details on numerical imple-
`mentation of viscoelastic models are given by Osswald and Hernández-Ortiz [37].
`As an example of the application of differential models to predict flow of polymeric
`liquids, it is worth mentioning the work by Dietsche and Dooley [38], who evalu-
`ated the White-Metzner, the Phan-Thien Tanner-1, and the Giesekus models by
`comparing finite element9 and experimental results of the flow inside multi-lay-
`ered coextrusion dies. Figure 5.23 [39] presents the progression of a matrix of
`colored circular polystyrene strands flowing in an identical polystyrene matrix
`down a channel with a square cross section of 0.95 (cid:117) 0.95 cm. The cuts in the fig-
`ure are shown at intervals of 7.6 cm.
`The circulation pattern caused by the secondary normal stress differences inside
`non-circular dies were capt