- Email: [email protected]
Melt Rheology
127
CHAPTER 5
MELT RHEOLOGY
Rheology is the field of science that studies fluid behavior during flow-induced deformation. From the variety of materials that rheologists study, polymers have been found to be the most interesting and complex. Polymer melts are shear thinning, viscoelastic, and their flow properties are temperature dependent. Viscosity is the most widely used material parameter when determining the behavior of polymers during processing. Since the majority of polymer processes are shear-rate dominated, the viscosity of the melt is commonly measured using shear deformation measurement devices. However, there are polymer processes, such as blow molding, thermoforming, and fiber spinning, that are dominated by either elongational deformation or by a combination of shear and elongational deformation. In addition, some polymer melts exhibit significant elastic effects during deformation. Rheometry is the aspect of rheology that measures by means of analytical techniques the rheological properties required for process layout and assessment. Furthermore, rheological characterization is an important tool for studying the molecular weight distribution of polymers, as well as for troubleshooting, optimizing, and designing processing equipment. Some industrial applications and applications are discussed later in this chapter.
128
5 Melt Rheology
5.1 BASIC CONCEPTS AND TERMINOLOGY This section summarizes the basic concepts and terminology commonly used in the field of rheology. Stress: Stress, τ , is the amount of force applied in a particular area. Depending of the type of flow, it is common to define the following stresses: shear stress and normal stress. Strain: Strain, γ, quantifies the deformation imposed on a material element. Depending of the type of flow, we distinguish between shear strains and elongational strains. Strain rate: The strain strain rate, γ, ˙ is sometimes also referred to as a rate of deformation: or shear rate. The natural response of a solid is a finite strain; however, for a liquid, the natural response is a strain rate rather than a finite strain. Viscosity: Viscosity, η, is the rheological property that relates the stress to the strain rate. In a Newtonian fluid, the deviatoric stresses that occur during deformation, τ , are directly proportional to the rate of deformation tensor, γ, ˙ τ = μγ. ˙
(5.1)
For Newtonian liquids, the viscosity, μ, is considered to be only dependent on temperature. However, the viscosity of most polymer melts is shear thinning in addition to being temperature dependent. The shear thinning effect is the reduction in viscosity at high rates of deformation. This phenomenon occurs because, at high rates of deformation, the molecules are stretched out and disentangled, enabling them to slide past each other with more ease, hence lowering the bulk viscosity of the melt. To take into consideration these non-Newtonian effects, it is common to use a viscosity which is a function of the strain rate and temperature to calculate the stress tensor in Eq. 5.1 τ = η(γ, ˙ T )γ, ˙
(5.2)
where η is the non-Newtonian viscosity and γ˙ the magnitude of strain rate or rate of deformation tensor defined by 1 γ˙ = II, (5.3) 2 where II is the second invariant of the strain rate tensor defined by γ˙ ij γ˙ ji . (5.4) II = i
j
The strain rate tensor components in Eq. 5.4 are defined by γ˙ ij =
∂ui ∂uj + . ∂xj ∂xi
(5.5)
The temperature dependence of the polymer’s viscosity is normally factored out as η(γ, ˙ T ) = f (T )η(γ), ˙
(5.6)
5.1 Basic Concepts and Terminology
Figure 5.1:
129
Viscosity curve for a polypropylene
100004 10 Compression molding Calendering
Injection molding
Extrusion
Fiber spinning
220 oC ABS
10003 10
260 oC 300 oC PC 340 oC 260 oC PA6
1002 10 300 oC
101 10
10
100
1000
10000
100000
Shear rate γ˙
Figure 5.2:
Viscosity curves for a selected number of thermoplastics
where for small variations in temperature, f (T ) can be approximated using an exponential function such as f (T ) = e-a(T -T0 ) .
(5.7)
Figure 5.1 illustrates the typical viscosity curve of a polymeric melt at three different temperatures. As presented in this figure, the viscosity curve is best presented in double-log graphs. Figures 5.2 through 5.6 show the shear thinning behavior and temperature dependence of the viscosity for a selected number of thermoplastics. Elongational viscosity: In polymer processes such as fiber spinning, blow molding, thermoforming, foaming, certain extrusion die flows, and compression molding with specific processing conditions, the major mode of deformation is elongational. To illustrate elongational flows, consider the fiber-spinning process shown in Fig. 5.7.
130
5 Melt Rheology
100000 PE-HD Low MFI (230 oC)
Pa•s
10000
1000 PP Low MFI (230 oC) PP High MFI (230 oC)
100
PE-HD High MFI (230 oC)
10 1
10
100
1000
10000
s -1
100000
Rate of deformation
Figure 5.3:
Viscosity curves for PE-HD and PP with low and high MFI at 230 ◦ C
10000 Pa•s
SAN (250 oC) ABS (250 oC)
PBT (250 oC)
1000
o
PS (250 C)
PA6 (250 oC) 100
10
1 1
10
100
1000
10000
s -1
Rate of deformation
Figure 5.4:
Viscosity curves for a selected number of thermoplastics at 250 ◦ C
100000
5.1 Basic Concepts and Terminology
131
10000 ASA (240 oC) Pa•s PES (340 oC)
1000
SB (240 oC)
PA66 (290 oC)
100
10
1 1
10
100
1000
s -1
10000
100000
Rate of deformation
Figure 5.5:
Viscosity curves for selected thermoplastics
100000
PMMA Low MFI (200 oC)
Pa•s 10000
PMMA High MFI (200 oC) 1000
PMMA High MFI (220 oC)
PAEK (400 oC) PMMA Low MFI (240 oC)
100
LCP (340 oC) 10
1 1
10
100
1000
10000
s -1
Rate of deformation
Figure 5.6:
Viscosity curves for PMMA, PAEK, and LCP
F
Figure 5.7:
Schematic diagram of a fiber-spinning process
100000
132
5 Melt Rheology
A simple elongational flow is developed as the filament is stretched with the following components of the rate of deformation: γ˙ 11 = - ˙ γ˙ 22 = - ˙
(5.8)
γ˙ 33 = 2 ˙ where ˙ is the elongation rate, and the off-diagonal terms of γ˙ ij are all zero. The diagonal terms of the total stress tensor can be written as σ11 = -p-η ˙ σ22 = -p-η ˙ σ33 = -p+2η ˙
(5.9)
Since the only outside forces acting on the fiber are in the axial or 3-direction, for the Newtonian case, σ11 and σ22 must be zero. Hence,
and
p = -η ˙
(5.10)
σ33 = 3η ˙ = η¯ , ˙
(5.11)
which is known as elongational viscosity or Trouton viscosity [1]. This is analogous to elasticity where the following relation between elastic modulus, E, and shear modulus, G, can be written E = 2(1+ν), (5.12) G where ν is Poisson’s ratio. For the incompressibility case, where ν = 0.5, Eq. 5.12 reduces to E = 3. (5.13) G 5•108
Viscosity (Pa-s)
108
Elongational test
μo = 1.6•108 Pa-s ηo = 5.5•107 Pa-s
5•107 ηo = 5•10 7 Pa-s
Shear test
107 5•106
106 102
Figure 5.8:
μo = 1.7•108 Pa-s
T = 140 °C Polystyrene I Polystyrene II 5•102 103
104 Stress, τ, σ (Pa)
105
5•105
Shear and elongational viscosity curves for two types of polystyrene
5.1 Basic Concepts and Terminology
133
6 LDPE
Log(Viscosity) (Pa-s)
5 Ethylene-propylene copolymer
4
PMMA POM
3
PA66
2 3
4
5
6
Tensile stress , log σ (Pa)
Figure 5.9:
Elongational viscosity curves as a function of tensile stress for several thermoplastics 0.8 0.7
60 °C
Degree of cure, c
0.6 50 °C
0.5 0.4
40 °C 0.3 0.2 0.1 0.0 10 -1
Figure 5.10: temperatures
10 0 10 1 Cure time (min)
10 2
Degree of cure as a function of time for a vinyl ester at various isothermal cure
Figure 5.8 [2] shows shear and elongational viscosities for two types of polystyrene. In the region of the Newtonian plateau, the limit of 3, shown in Eq. 5.11, can be seen. Figure 5.9 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 difficult task. For example, to maintain a constant strain rate, the specimen must be deformed both uniformly and exponentially. In addition, a molten polymer must be tested completely submerged in a heated, neutrally buoyant liquid at a constant temperature.
134
5 Melt Rheology
10 60 °C 50 °C
Viscosity (Pa-s)
40 °C
100
10 -1 0
Figure 5.11: temperatures
0.1
0.2 0.3 0.4 0.6 Degree of cure, c
0.5
0.7
Viscosity as a function of degree of cure for a vinyl ester at various isothermal cure
Curing effects of viscosity A curing thermoset polymer has a conversion or cure dependent viscosity that increases as the molecular weight of the reacting polymer increases. For vinyl ester, whose curing history is shown in Fig. 5.10 [3], the viscosity behaves as shown in Fig. 5.11 [3]. Hence, a complete model for viscosity of a reacting polymer must contain the effects of strain rate, γ, ˙ temperature, T , and degree of cure, c, such as η = η(γ, ˙ T, c).
(5.14)
There are no generalized models that include all these variables for thermosetting polymers. However, extensive work has been done on the viscosity of polyurethanes [4, 5] used in the reaction injection molding process. An empirical relation that models the viscosity of these mixing-activated polymers, given as a function of temperature and degree of cure, is written as η = η0 eE/RT
cg cg − c
c1 +c2 c (5.15)
where E is the activation energy of the polymer, R is the ideal gas constant, T is the temperature, c g is the gel point1 , c the degree of cure, and c 1 and c2 are constants that fit the experimental data. Figure 5.12 shows the viscosity as a function of time and temperature for a 47% MDI-BDO P(PO-EO) polyurethane. 1 At the gel point, the cross-linking creates a closed network, at which point it is said that the molecular weight goes to infinity.
5.1 Basic Concepts and Terminology
135
100 90 °C 50 °C 30 °C
Viscosity (Pa-s)
10
1
0.1
0.01 0.1
1.0
10
100
Time (min)
Figure 5.12: Viscosity as a function of time for a 47% MDI-BDO P(PO-EO) polyurethane at various isothermal cure temperatures
Suspension rheology: Particles suspended in a material, such as in filled or reinforced polymers, have a direct effect on the properties of the final article and on the viscosity during processing. Numerous models have been proposed to estimate the viscosity of filled liquids [6, 7, 8, 9, 10]. Most models proposed are a power series of the form ηf = 1+a1 φ+a2 φ2 +a3 φ3 +..... η
(5.16)
The linear term in Eq. 5.16 represents the narrowing of the flow passage caused by the filler that is passively entrained by the fluid and sustains no deformation as shown in Fig. 5.13. For Particles V
V v(z)
z . γ
Figure 5.13:
. κγ
vf (z)
Schematic diagram of strain rate increase in a filled system
instance, Einstein’s model, which only includes the linear term with a 1 = 2.5, was derived based on a viscous dissipation balance. The quadratic term in the equation represents the first-order effects of interaction between the filler particles. Geisb¨usch suggested a model with a yield stress and where the strain rate of the melt increases by a factor κ as ηf =
τ0 + κη0 (κγ). ˙ γ˙
(5.17)
136
5 Melt Rheology
7 Experimental data a1 = 2.5, a = 14.1 (Guth, 1938)
6
2
f
η /η
5 4 3 2 1 0
10 30 40 20 Volume fraction of filler (%)
50
Figure 5.14: Viscosity increase as a function of volume fraction of filler for polystyrene and lowdensity polyethylene containing spherical glass particles with diameters ranging between 36 μm and 99.8 μm
For high deformation stresses, which are typical in polymer processing, the yield stress in the filled polymer melt can be neglected. Figure 5.14 compares Geisb¨usch’s experimental data to Eq. 5.16 using the coefficients derived by Guth [9]. The data and Guth’s model seem to agree well. A comprehensive survey on particulate suspensions was recently given by Gupta [11] and on short-fiber suspensions by Milliken and Powell [12]. Complex modulus: The complex modulus G ∗ is the overall resistance to deformation of a material, regardless of whether that deformation is recoverable (elastic) or non-recoverable (viscous). Complex viscosity: The complex viscosity is a ratio of complex modulus to angular frequency, usually denoted by the symbol η ∗ . Dynamic storage modulus: The dynamic storage modulus is the contribution of elastic (solid-like) behavior to the complex modulus, usually denoted by the symbol G . Dynamic loss modulus: The dynamic loss modulus is the component of the complex modulus that is out of face, usually denoted by the symbol G . Compliance: The compliance, J, is the ratio of strain to stress. Viscoelastic memory effects: When a polymer melt is deformed, either by stretching, shearing, or often by a combination of the above, the polymer molecules are stretched and untangled. In time, the molecules try to recover their initial shape, in essence getting used to their new state of deformation. If the deformation is maintained for a short period of time, the molecules may go back to their initial position, and the shape of the melt is fully restored to its initial shape. Here, it is said that the molecules remembered their initial position.
5.1 Basic Concepts and Terminology
137
However, if the shearing or stretching goes on for an extended period of time, the polymer cannot recover its starting shape, in essence forgetting the initial positions of the molecules. The time it takes for a molecule to fully relax and get used to its new state of deformation is referred to as the relaxation time, λ. A useful parameter often used to estimate the elastic effects during flow is the Deborah number 2, De. The Deborah number is defined by De =
λ , tprocess
(5.18)
where tprocess is a characteristic process time. For example, in an extrusion die, a characteristic process time can be defined by the ratio of characteristic die dimension in the flow direction and average speed through the die. A Deborah number of zero represents a viscous fluid 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 extrudate dimension deviations or irregularities such as extrudate swell, shark skin, or even melt fracture. Melt fracture: This phenomenon appears in the form of waves in the extrudate when the polymer is extruded at high speeds and is not allowed to relax. a)
b)
c)
d)
Figure 5.15:
Various shapes of extrudates under melt fracture [14]
This phenomenon is sometimes referred to as shark skin and is shown for a high-density polyethylene in Fig. 5.15a [13]. It is possible to extrude at such high speeds that an intermittent separation of melt and inner die walls occurs, as shown in Fig. 5.15b. This phenomenon is often referred to as the stick-slip effect or spurt flow 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. If the speed is further increased, a helical geometry is extruded as 2 From the Song of Deborah, Judges 5:5 – "The mountains flowed before the Lord." M. Rainer is credited for naming the Deborah number; Physics Today, 1, (1964).
138
5 Melt Rheology
shown for a polypropylene extrudate in Fig. 5.15c. Eventually, the speeds are so high that a chaotic pattern develops, such as the one shown in Fig. 5.15d. This well-known phenomenon is called melt fracture. The shark skin effect is frequently absent, and spurt flow seems to occur only with linear polymers. Table 5.1 presents estimated critical melt fracture stresses for various polymers. Table 5.1:
Critical shear stress for flow instabilities prediction for some polymers Material
Critical shear stress, (MPa)
HDPE
0.145
LLDPE
0.145
LDPE
0.130
PP
0.130
PS
0.130
Normal stresses: 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 τxy = η(γ) ˙ γ˙ xy . (5.19) Measurable normal stress differences, N 1 = τxx -τyy and N2 = τyy -τzz are referred to as the first and second normal stress differences.
Reduced first normal stress difference Log(ψ /a T2 ) (Pa-s 2 )
7 Experiments done between o115 and 210oC 115 C 130 o C 150 o C 170 o C 190 o C 210 o C
6
5
4
3
2
1 -4
-3
-2
-1
0
1
2
3
4
. Reduced shear rate, log aT γo (s-1)
Figure 5.16: Reduced first normal stress difference coefficient for a low-density polyethylene melt at a reference temperature of 150 ◦ C
The first and second normal stress differences are material dependent and are defined by 2 N1 = τxx -τyy = -Ψ1 γ˙ xy
N2 = τyy -τzz =
2 -Ψ2 γ˙ xy .
(5.20) (5.21)
5.1 Basic Concepts and Terminology
139
Reduced viscosity, Log(η/aT) (Pa-s)
7 115 o C 130 o C 150 o C 170 o C 190 o C 210 o C
6
5
4
3
2 1 -4
Figure 5.17: 150 ◦ C
-3
-2
-1 0 1 2 . Reduced shear rate, log aT γo (s-1)
3
4
Reduced viscosity for a low-density polyethylene melt at a reference temperature of
The material functions, Ψ 1 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. 5.20 and 5.21. Figure 5.16 presents the first normal stress difference coefficient for the low density polyethylene melt of Fig. 5.17 at a reference temperature of 150 ◦ C. The second normal stress difference is much smaller than the first normal stress difference and is therefore difficult to measure. Material data banks: The rheological behavior of molten polymers are found in material data banks, such as CAMPUS , and are presented in graphs such as shown in Figs. 5.16 and 5.17. Surface tension: Surface tension plays a significant role in the deformation of polymers during flow, especially in dispersive mixing of polymer blends. Surface tension, σ S , between two materials appears as a result of different intermolecular interactions. In a liquid-liquid system, surface tension manifests itself as a force that tends to maintain the surface between the two materials to a minimum. Thus, the equilibrium shape of a droplet inside a matrix, which is at rest, is a sphere. When three phases touch, such as liquid, gas, and solid, we get different contact angles depending on the surface tension between the three phases. Figure 5.18 schematically depicts three different cases. In case 1, the liquid perfectly wets the surface with a continuous spread, leading to a wetting angle of zero. Case 2, with moderate surface tension effects, shows a liquid that has a tendency to flow over the surface with a contact angle between zero and π/2. Case 3, with a high surface tension effect, is where the liquid does not wet the surface, which results in a contact angle greater than π/2. In Fig. 5.18, σ S denotes the surface tension between the gas and the solid, σ l the surface tension between the liquid and the gas, and σ sl the surface tension between the solid and liquid. The wetting angle can be measured using simple techniques such as a projector, as shown schematically in Fig. 5.19. This technique, originally developed by Zisman, can be used in
140
5 Melt Rheology
Case 1
σl
Case 2
σs
φ
σs,l
Case 3
Figure 5.18: effects
Schematic diagram of contact between liquids and solids with various surface tension
φ Micrometer Syringe Drop Magnifying apparatus xy-translator Optical bench
Figure 5.19:
Schematic diagram of apparatus to measure contact angle between liquids and solids
cosφ
1.0
0.5
0
Figure 5.20:
σc
σl
Contact angle as a function of surface tension
the ASTM D2578 standard test. Here, surface tension, σ l is applied to a film. The measured values of cos φ are plotted as a function of surface tension σ l , as shown in Fig. 5.20, and extrapolated to find the critical surface tension, σ c , required for wetting. For liquids of low-viscosity, a useful measurement technique is the tensiometer, schematically represented in Fig. 5.21. Here, the surface tension is related to the force it takes to pull a platinum ring from a solution. Surface tension for selected polymers are listed in Table 5.2, for some solvents in Table 5.3, and between polymer-polymer systems in Table 5.4. There are many areas in polymer processing and in engineering design with polymers where surface tension plays a significant role. These areas are the mixing of polymer blends, adhesion, treatment of surfaces to make them non-adhesive, and sintering. During manufacturing, it is often necessary to coat and crosslink a surface with a liquid adhesive or bonding
5.1 Basic Concepts and Terminology
141
material. To enhance adhesion it is often necessary to raise surface tension by oxidizing the surface, by creating COOH-groups, using flames, etching, or releasing electrical discharges. Table 5.2:
Typical surface tension values of selected polymers at 180 ◦ C Polymer
σs (N/m)
∂σs /∂T (N/m/K)
0.0290
-
0.035
-5.7 × 10-5
Polyethylene teraphthalate (290 C)
0.045
-6.5 × 10-5
Polyisobutylene
0.0234
-6 × 10-5
Polymethyl methacrylate
0.0289
-7.6 × 10-5
Polypropylene
0.030
-5.8 × 10-5
Polystyrene
0.0292
-7.2 × 10-5
Polytetrafluoroethylene
0.0094
-6.2 × 10-5
Polyamide resins (290 ◦ C) Polyethylene (linear) ◦
Level Force measuring device
Ring
Fluid
Figure 5.21:
Schematic diagram of a tensiometer used to measure surface tension of liquids
Table 5.3:
Surface tension for several solvents Solvent n-Hexane
Surface tension - σs (N/m) 0.0184
Formamide
0.0582
Glycerin
0.0634
Water
0.0728
142
5 Melt Rheology
Table 5.4:
Surface tension between polymers Polymers PE-PP
σs (N/m)
∂σs /∂T (N/m/K)
T (◦ C)
0.0011
-
140 -5
PE-PS
0.0051
2.0×10
180
PE-PMMA
0.0090
1.8 × 10-5
180
PP-PS
0.0051
-
140
PS-PMMA
0.0016
1.3 × 10-5
140
This is also the case when enhancing the adhesion properties of a surface before painting. On the other hand, it is often necessary to reduce adhesiveness of a surface, such as required when releasing a product from the mold cavity or when coating a pan to give it nonstick properties. A material often used for this purpose is polytetrafluoroethylene (PTFE), mostly known by its tradename Teflon. 5.2 CONSTITUTIVE MODELS A constitutive model is a mathematical expression that relates stress, strain, stress rate, strain rate, pressure, and temperature, just to name a few. The most common constitutive model relates stress to strain rate and temperature. The relationship between stress and strain rate is a viscosity function that typically incorporates the dependence of other factors, such as temperature, pressure, and time. It can be stated that the viscosity is a complex function that depends on factors including: • Macromolecular characteristic of the polymer, such as chain branches and rigidity • Molecular weight distribution • Shear rate • Temperature • Pressure • Time • Voltage in the case of electro-rheological fluids • Magnetic field in case of magneto-rheological fluids Although several viscosity functions are presented in the literature, in this book the Newtonian model, the Power law and Bird-Carreau-Yasuda model were selected because they can be easily related to most polymer flows that involve shear. For non-isothermal and nonisobaric problems, models that relate the viscosity to temperature and pressure were also included. Finally, for complex flows, such as those that involve viscoelastic and elongational effects, the Phan-Thien and Tanner multimode model is presented.
5.2 Constitutive Models
143
5.2.1 Newtonian Model The Newtonian model was developed by Newton in 1687 and predicts a viscosity independent of the shear rate (i.e., the shear stress is a linear relationship of shear rate). Although this model is easy to apply, it can be used only for the beginning of the viscosity curve at low shear rates. Mathematically, the Newtonian model can be expressed as τ , γ˙
μ=
(5.22)
where μ is the shear viscosity (Pa· s), γ˙ is the shear rate (s -1 ), and τ is the stress (Pa). Note that for Newtonian fluids the viscosity is represented by μ. 5.2.2 Power Law Model This model, also known as the Ostwald and de Waale model, is easy to use and can represent the pseudoplastic behavior of polymer melts. The term pseudoplastic was introduced by Williamson and Ostwald in 1925 to describe the behavior of the fluids that suffer a reduction of viscosity with an increase of shear rate. It can be applied for the pseudoplastic or shear thinning region because for low shear rates the prediction of viscosity is very high. Mathematically, the Power law model can be expressed as η = m · γ˙ n-1 ,
(5.23)
where η is the shear viscosity (Pa·s), γ˙ is the shear rate (s -1 ), m is the consistency factor (P a · sn ), and n is the Power law index. For polymers the Power law index n is typically between 0.2 and 0.9. The deviation of n from the unity is taken as a measurement of the non-Newtonian behavior. For values of n > 1, the fluid behaves as a dilatant, or shear thickening fluid. Table 5.5 presents a list of typical Power law and consistency indices for common thermoplastics. Table 5.5:
Power law and consistency indices for common thermoplastics
Polymer
m (Pa· sn )
n
T (◦ C)
High-density polyethylene
2.0 × 104
0.41
180
Low-density polyethylene
3
6.0 × 10
0.39
160
Polyamide 66
6.0 × 102
0.66
290
Polycarbonate
2
6.0 × 10
0.98
300
Polypropylene
7.5 × 103
0.38
200
Polystyrene
4
2.8 × 10
0.28
170
Polyvinyl chloride
1.7 × 104
0.26
180
The temperature dependence of the Power-law viscosity can be built into the consistency as m = m0 e-a(T -Tref )
(5.24)
144
5 Melt Rheology
or by introducing a time-temperature superposition shift factor, ˙ n-1 , η = k · aT · (aT · γ)
(5.25)
where for amorphous polymers, the shift factor is best defined using the Williams, Landel, and Ferry model 3 given by aT = 10
C1 (T -Tref ) 2 +(T -Tref )
-C
,
(5.26)
where C1 , C2 are empirical constants obtained by statistical regression of viscosity curve at different temperatures and selecting a particular value of reference temperature (C 1 = 17.44 and C2 = 51.6 when Tref = Tg . If Tref = Tg +45K, C1 = 8.86 and C2 = 101.6), T is the temperature (K), and T ref is the reference temperature (K). A shift commonly used for semicrystalline polymers is the Arrhenius shift, which is written as E0 1 1 , (5.27) Ln(aT ) = R T T0 where E0 is the activation energy (J· mol -1 ) (this constant can be obtained by statistical regression of the viscosity curve at different temperatures and selecting a particular value of reference temperature), T 0 is the reference temperature (K), and R is the universal gas constant (8.314 J· mol -1 · K-1 ).
5.2.3
Bird-Carreau-Yasuda Model
The Bird-Carreau-Yasuda model is more complex and has the advantage of predicting both the Newtonian and the pseudoplastic behavior of polymers, as well as the transition region. Mathematically, the Bird-Carreau-Yasuda model can be expressed as follows η − η0 = [1+|λγ| ˙ a ](n-1)/a , η0 − η∞
(5.28)
where η is the shear viscosity (Pa·s), η 0 is the zero-shear-rate viscosity (Pa·s), η ∞ is the infinite-shear-rate viscosity (Pa·s), λ is the time constant , γ˙ is the shear rate, (s -1 ), and n is the Power law index. In the original Bird-Carreau model, the constant a = 2. In many cases, the infinite-shearrate viscosity is negligible, reducing Eq. 5.28 to a three parameter model. Equation 5.28 was modified by Menges, Wortberg, and Michaeli to include a temperature dependence using a WLF relation. The modified model, which is used in commercial polymer data banks, is written as: A · aT η= , (5.29) [1+B γa ˙ T ]C 3 Often
referred to as WLF equation.
5.2 Constitutive Models
145
where aT is the temperature shift factor (A) is the viscosity constant (Pa·s -C ), B is the time constant (s), and C is the shear thinning index. The shift a T , for the best-fit reference temperature, T ref , is given by4 Log(aT ) =
8.86(Tref -Tg ) 8.86(T -Tg ) − , 101.6+Tref -Tg 101.6+T -Tg
(5.30)
where aT is the temperature shift factor, T g is the glass transition temperature (K), and T ref is the reference temperature (T ref = Tg + 45K). Table 5.6 presents constants for CarreauWLF (amorphous) and Carreau-Arrhenius models (semi-crystalline) for various common thermoplastics. Table 5.6: Constants for Carreau-WLF (amorphous) and Ccarreau-arrhenius (semicrystalline) models for various common thermoplastics Polymer
A (Pa·s)
B (s)
C
Tref (◦ C)
Tg (◦ C)
T0 (◦ C)
E0 (J/mol)
High-density polyethylene
24,198
1.38
0.60
-
-
200
22,272
Low-density polyethylene
317
0.015
0.61
-
-
189
43,694
Polyamide 66
44
0.00059
0.40
-
-
300
123,058
Polycarbonate
305
0.00046
0.48
320
153
-
-
Polypropylene
1,386
0.091
0.68
-
-
220
427,198
Polystyrene
1,777
0.064
0.73
200
123
-
-
Polyvinyl chloride
1,786
0.054
0.73
185
88
-
-
5.2.4 Pressure Dependence of Viscosity The viscosity of a polymer melt is also dependent on the pressure, which is particularly important in injection molding processes of thin parts where the pressure levels are the highest. The pressure dependence of viscosity can be modeled by using: • Pressure Coefficient – The viscosity at a certain pressure can be predicted based on a viscosity data at another pressure if the coefficient of pressure is known, η(P ) = η(Po ) · eφ·P ,
(5.31)
where η(P ) is the viscosity at a pressure p (Pa· s), φ is the pressure coefficient atm-1 , and η(Po ) is the viscosity at a pressure p o (Pa· s). The pressure coeffcient can be obtained by statistical regression of viscosity data at various pressures. Very little information on the coefficient of pressure has been published so far. Table 5.7 presents the reported data for some particular polymers. 4 This
shift should be used for amorphous polymers.
146
5 Melt Rheology
Table 5.7:
Coefficient of pressure for some particular polymers
Polymer
Temperature, ◦ C
φ, 10-3 atm
180 190 200
6.00 4.00 3.00
-
2.31
PVC for injection molding
Linear low-density Polyethylene (LLDPE)
• Modified WLF model – In addition to the temperature shift, Menges, Wortberg, and Michaeli measured a pressure dependence of the viscosity and proposed the following model, which includes both temperature and pressure viscosity shifts: log η(T, p) − log η0 =
8.86(T -T0 ) 8.86(T -T0 +0.02p) , − 101.6+T -T0 101.6+(T -T0 +0.02p)
(5.32)
where η is the pressure dependent viscosity, η 0 is the reference viscosity, and p is the pressure in bar (the constant 0.02 represents a 2 ◦ C shift per bar). 5.2.5 Phan-Thien and Tanner Multimode Model A great variety of viscoelastic models have been used to describe the viscoelastic behavior of polymer melts, including the model independently developed by Phan-Thien and Tanner (PTT) [15] and by Acierno et al. [16]. The multimode Phan-Thien and Tanner model has been successfully used for modeling the shear, elongational, and oscillatory shear behavior of polymers, as well as for modeling the processing of viscoelastic polymers. Mathematically, the multimode Phan-Thien and Tanner (PTT) 2D viscoelastic model in cylindrical coordinates, can be expressed as
exp(
i λi dτzzi dV dV − 2λi (1 − ξi )τzzi = 2ηi (τzzi +2τrri ))τzzi +λi V ηi dz dz dz
(5.33)
i λi dV dV dτrri +λi (1 − ξi )τrri = −ηi (τzzi +2τrri ))τrri +λi V ηi dz dz dz
(5.34)
exp(
τzz =
N i=1
τzzi
τrr =
N
τrri
(5.35)
i=1
where τzz is the z component of the elongational stress, τ rr is the r component of the elongational stress, V is the velocity in z direction, η i is the elongational viscosity parameter for PTT model, λ i is the relaxation time parameter for PTT model, i is the parameter for PTT model, and ξ i is the parameter for PTT model.
5.3 Rheometry
147
5.3 RHEOMETRY There are various ways to qualify and quantify the properties of the polymer melt in industry. The techniques range from simple analyses for checking the consistency of the material at certain conditions to more complex measurements that evaluate viscosity and normal stress differences. This section includes three such techniques to give the reader a general idea of current measuring methods. 5.3.1 The Melt Flow Indexer The melt flow indexer is often used in industry to characterize a polymer melt and as a simple and quick means of quality control. It takes a single point measurement using standard testing conditions specific to each polymer class on a ram-type extruder or extrusion plastometer as shown in Fig. 5.22. Weight
Thermometer
Polymer
Capillary
Figure 5.22:
Schematic diagram of an extrusion plastometer used to measure the melt flow index
The standard procedure for testing the flow rate of thermoplastics using an extrusion plastometer is described in the ASTM D1238 test as presented in Table 5.8. During the test, a sample is heated in the barrel and extruded from a short cylindrical die using a piston actuated by a weight. The weight of the polymer in grams extruded during the 10-minute test is the melt flow index (MFI) of the polymer.
148
5 Melt Rheology
Table 5.8: Standard methods of measuring melt flow index (MFI): melt flow rate (MFR), melt volume rate (MVR), and flow rate ratio (FRR)(after Shastri) Standard
ISO 1133
ASTM D1238 - 98
Specimen
Powder, pellets, granules, or strips of films
Powder, pellets, granules, strips of films, or molded slugs
Conditioning
In accordance with the material standard, if necessary
Check the applicable material specification for any conditioning requirements before using this test. See practice D618 for appropriate conditioning practices.
Apparatus
Extrusion plastometer with a steel cylinder (115 – 180) mm (L) x 9.55 ±0.025 mm (D), and a die with an orifice of 8.000 ±0.025 mm (L) x 2.095 ±0.005 mm (D)
Extrusion plastometer with a steel cylinder 162 mm (L) x 9.55 ±0. 008 mm (D), and a die with an orifice of 8.000 ±0.025 mm (L) x 2.0955 ±0.0051 mm (D)
Test procedures
Test temperature and test load as specified in Part 2 of the material designation standards, or as listed in ISO 1133. Some examples are: PC (300 ◦ C /1.2 kg) ABS (220 ◦ C /10 kg) PS (200 ◦ C /5 kg) PS-HI (200 ◦ C /5 kg) SAN (220 ◦ C /10 kg) PP (230 ◦ C /2.16 kg) PE (190 ◦ C / 2.16 kg) POM (190 ◦ C /2.16 kg) PMMA (230 ◦ C /3.8 kg) Charge ⇒ within 1 min Preheat ⇒ 4 min Test time ⇒ last measurement not to exceed 25 min from charging. Procedure A – manual operation using the mass and cuttime intervals shown in the following:
Test temperature and test load as specified in the applicable material specification, or as listed in D1238. Some examples are: PC (300 ◦ C /1.2 kg) ABS (230 ◦ C /10 kg) PS (200 ◦ C /5 kg) PS-HI (200 ◦ C /5 kg) SAN (220 ◦ C /10 kg) PP (230 ◦ C /2.16 kg) PE (190 ◦ C / 2.16 kg) POM (190 ◦ C /2.16 kg) Acrylics (230 ◦ C /3.8 kg) Charge ⇒ within 1 min Preheat ⇒ 6.5 min Test time ⇒ 7.0 ±0.5 min from initial charging. Procedure A – manual operation using the mass and cuttime intervals shown in the following: Continued on next page
5.3 Rheometry
149
Standard
Values and units
ISO 1133
ASTM D1238 - 98
MFR; Mass; Time 0.1 to 0.5g/10min; 3–5g ; 4min >0.5 to 1g/10min; 4–5g; 2min >1 to 3.5g/10min; 4–5g; 1min >3.5 to 10 g/10 min 6– 8g; 30s > 10g/10min; 6–8g; 5-15s Procedure B – automated time or travel indicator is used to calculate the MFR (MVR) using the mass as specified above in Procedure A for the predicted MFR
MFR; Mass; Time 0.15 to 1g/10min; 2.5–3g; 6 min >1 to 3.5g/10min; 3–5g; 3 min >3.5 to 10g/10min; 4–8g; 1 min >10 to 25g/10min; 4–8g; 30s > 25g/10min; 4–8g; 15s Procedure B – MFR (MVR) is calculated from automated time measurement based on specified travel distances, < 10 MFR ⇒ 6.35 ±0.25 mm. > 10 MFR ⇒ 25.4 ±0.25 mm. and using the mass as specified above for the predicted MFR
MFR ⇒g/10min MVR ⇒cm3 /10min
MFR ⇒ g/10min MVR ⇒ cm3 /10min FRR ⇒ Ratio of the MFR (190/10) by MFR (190/2.16) (used specifically for PE)
INDUSTRIAL APPLICATION 5.1
Industrial Application of Weathering of an Exterior Polyethylene Application A failed polyethylene component exposed to the elements was analyzed to determine the cause of failure (Figure 5.23). Most likely, the cause of failure was degradation due to UV radiation. UV rays can lead to molecular chain sission, often leading to a significant reduction in molecular weight.
Figure 5.23:
Failed polyethylene part
150
5 Melt Rheology
Abs
Part surface
Abs
Part core
Abs
Polyethylene 0.6 0.4 0.2 4000
3000
2000
-1
Wavelength (cm )
1000
500
Figure 5.24: FTIR Spectrums of the surface of the failed part (top), inner core (center) and reference polyethylene (bottom)
A common test performed to detect a loss in molecular weight is measuring changes in melt flow index. For the failed parts the melt flow index was measured to be above 150 g/10 minutes. The material specified for this application was an HDPE with a melt flow index of 15 g/10 minutes. This large difference between specified and tested MFI is due to a significant loss in molecular weight as a result of UV degradation. Another analytical test that can be performed to detect UV degradation is a Fourier Transform Infrared Spectroscopy (FTIR). FTIR spectroscopy was performed at the surface and core of the HDPE sample, and is shown in Fig. 5.24. The FTIR shows the typical spectral results expected for polyethylene. However, the spectrum shows two additional absorption bands (one between 1750 cm -1 and 1700 cm -1 , and another between 1300 cm -1 and 1100 cm -1 ) that indicate the formation of carbonyls and byproducts associated with oxidation, a result of UV degradation. The FTIR performed at the surface of the part shows stronger absorption bands compared to the FTIR at the core. Therefore, as expected, the level of oxidation at the surface is much higher than the oxidation at the core of the part. If the reduction in properties had been caused by oxidation during processing the part would have exhibited uniform degradation throughout the thickness of the part. 5.3.2 Capillary Viscometer The most common and simplest device for measuring viscosity is the capillary viscometer. Its main component is a straight tube or capillary, and it was first used to measure the viscosity of water by Hagen and Poiseuille. A capillary viscometer has a pressure driven flow for which the velocity gradient or strain rate and also the shear rate will be maximum at the wall and zero at the center of the flow, making it a non-homogeneous flow. Since
5.3 Rheometry
151
pressure driven viscometers employ non-homogeneous flows, they can only measure steady shear functions such as viscosity, η(γ). ˙ However, they are widely used because they are relatively inexpensive to build and simple to operate. Despite their simplicity, long capillary viscometers provide the most accurate viscosity data available. Another major advantage is that the capillary viscometer has no free surfaces in the test region, unlike other types of rheometers, such as the cone and plate rheometers, which we will discuss in the next section. When the strain rate dependent viscosity of polymer melts is measured, capillary viscometers may provide the only satisfactory method of obtaining such data at shear rates >10 s-1 . This is important for processes with higher rates of deformation such as mixing, extrusion, and injection molding. Other advantages of the capillary viscometer include. • Capillary flows and geometries are very similar to those encountered in real processing equipment. • A capillary viscometer can be adapted to on-line measurement. • The system allows the study of flow anomalies such as extrudate swell, melt fracture, or stick-slip conditions. • A capillary viscometer can be used to study the pressure dependence of viscosity Heater
Insulation
Pressure transducer
L Polymer sample
Extrudate R
Figure 5.25:
Schematic diagram of a capillary viscometer
As shown in Fig. 5.25, in a capillary viscometer the material is fed into a cylinder where the temperature is maintained within a very narrow range (about T ±0.5 ◦ C). Once the material is molten, the piston traveling at a well controlled speed pushes the material through the capillary. The pressure is measured at the inlet of the circular capillary, and for rectangular capillaries the pressure can be measured inside of the capillary. Using the piston speed, the dimensions of the piston, and the capillary, the apparent shear rate can be calculated; and using the pressure and the dimensions of the capillary, the apparent shear stress can be computed. With the apparent shear rate and shear stress, the apparent viscosity can be computed. Tests at different piston speeds and temperatures can be carried out to obtain an apparent viscosity curve. Subsequent corrections on shear rate and shear stress allow the prediction of the actual viscosity curve. The capillary extrudate can be collected to observe the range of flow anomalies when they occur.
152
5 Melt Rheology
The capillary viscometer includes the following instrumentation: • • • •
A piston and its speed control A cylinder with temperature control An auxiliary extruder and purge valve (optional) A capillary set
There are three types of capillaries in use today: • Circular capillary – This capillary works best for higher shear rates, so it is useful to define the pseudoplastic behavior of polymer melts. The geometrical parameters required to obtain the viscosity data are the inside diameter D (or radius R) and the length L. • Rectangular capillary – This capillary is used for lower shear rates, so it is useful to measure the Newtonian plateau of a polymer melt. This capillary has the advantage that the pressure can be measured directly inside the capillary. Therefore, inlet pressure correction, such as Bagley corrections, are not required. This type of capillary is easier to clean up. The required geometrical parameters to obtain the viscosity data are the internal height h, the width b, and the length L. To minimize the entrance effects, a b/h ratio of 15:1 is normally used. To obtain a complete viscosity curve, measurements with circular and rectangular capillaries are usually done. • Annular capillary – This capillary allows the alteration of the length and the radius by changing only the core and does not have entrance effects. The main disadvantages are that it is difficult to control the temperature of the core and that a considerably higher amount of polymer melt is required. The required geometrical parameters to obtain the viscosity data are the inside diameter D i (or inside radius R i ), the outside diameter Do (or outside radius R o ), and the length L. Annular capillaries are fabricated with a diameter of 20 mm and ratio D i /(Do − Di ) higher than 20. For accurate measurements a piston displacement control is needed. Three possible piston displacement controls exist. These are: • Constant speed control – This control provides a constant piston speed, if the seal between piston and cylinder is adequate. In this alternative the displacement of the piston is controlled while the pressure is registered. Two possible constant speed controllers can be used: – Mechanical – This controller uses a screw that moves axially at a constant linear speed – Hydraulic – This controller uses a very precise hydraulic system to push the piston at a carefully controlled speed • Constant pressure control – Here, pressurized gas pushes the piston. With the help of a pressure sensor located at the inlet (or inside) of the capillary, a constant pressure can be obtained and the volumetric flow can be measured. • Constant volume control – this technology uses a gear pump or an extruder to control the flow of melt while the pressure is measured. Although this alternative nears actual processing conditions, controlling a constant flow is difficult to attain.
5.3 Rheometry
153
The extruder is often a desirable and recommended component of the capillary viscometer to avoid air entrapment and to obtain thermal homogeneity in the melt. A manual load of the cylinder can be done. The extruder delivers the molten polymer to be fed to the cylinder by means of a three-way valve. Corrections: Because of geometrical constraints, the pressure sensor cannot be placed inside the circular capillary. The deformation at the inlet of the capillary leads to pressure losses that require a correction of the shear stress. The correction of the inlet pressure loss is normally called the Ryder-Bagley correction. In the calculation of the shear rate, Newtonian equations are used, so it is necessary to correct for the non-Newtonian behavior of polymer melts. The correction of the non-Newtonian behavior is called the WeissenbergRabinowitsch correction. Ryder-Bagley Correction: For the Ryder-Bagley correction, measurements of pressure for several capillary lengths must be done. At least three capillary lengths are normally used. When the sensor pressures plotted as a function of the capillary lengths at each shear rate, straight lines (or parabolic lines for higher shear rates) are obtained (see Fig. 5.26). The intercept with the y-axis is the Ryder-Bagley correction and has to be deducted from the measured pressure to obtain the real shear stress. The corrected shear stress for a circular capillary can be calculated using τ=
(ΔP − ΔPBagley ) · R , 2·L
(5.36)
where τ is the corrected shear stress (Pa), ΔP is the measured pressure (Pa), ΔP Bagley is the Ryder-Bagley correction (Pa), R is the capillary internal radius (mm), and L is the capillary length (mm).
Figure 5.26:
Ryder-Bagley correction for polypropylene at different shear rates
154
5 Melt Rheology
Table 5.9:
Coefficient of pressure for some particular polymers Circular capillary
Rectangular capillary
Shear rate for a Newtonian fluid 4·V˙ γ˙ ap = π·R 3
Shear rate for a Newtonian fluid 6·V˙ γ˙ ap = b·h 2
Weissenberg-Rabinowitsch correction
Weissenber-Rabinowitsch correction
First point (i = 1) γ˙ wap2 γ˙ w1 = 34 γ˙ wap1 + 14 τwap1 τwap
First point (i = 1) γ˙ 2 γ˙ w1 = 23 γ˙ wap1 + 13 τwap1 wap τw
Intermediate point γ˙ −γ˙ wapi-1 i+1 γ˙ w1 = 34 γ˙ wapi + 14 τw1 wap τw −τw
Intermediate point γ˙ −γ˙ wapi-1 i+1 γ˙ w1 = 23 γ˙ wapi + 13 τw1 wap τw -τw
Last point (i = n) ˙ γ˙ wn = γ˙ wn V˙Vn
Last point (i = n) ˙ γ˙ wn = γ˙ wn V˙Vn
2
i+1
i-1
2
n-1
i+1
i-1
n-1
Weissenberg-Rabinowitsch Correction: As mentioned earlier, Newtonian equations are used in the calculation of the shear rate. Because polymer melts are non-Newtonian, a correction must be done that takes into account the shear thinning behavior. This correction is the socalled Weissenberg-Rabinowitsch correction. Table 5.9 presents the equations to calculate the shear rate for Newtonian fluids and the Weissenberg-Rabinowitsch correction for circular and rectangular capillaries. The standardized techniques used to measure rheological properties of polymeric materials by means of a capillary viscometer are the ISO 11443 and the ASTM D3835 tests. Both tests are presented in Table 5.10. The ASTM D5099 test is used to measure rheological properties of rubber materials using capillary viscometry. The ASTM D5099 test is presented in Table 5.11. Table 5.10: Standard test method for determination of properties of polymeric materials by means of a capillary viscometer Standard
ISO 11443:1995
ASTM D3835-02
Abstract
ISO 11443:2005 specifies methods for determining the fluidity of polymer melts subjected to shear stresses at rates and temperatures approximating to those arising in plastics processing.
This test method covers measurement of the rheological properties of polymeric materials at various temperatures and shear rates common to processing equipment. Continued on next page
5.3 Rheometry
155
Standard
ISO 11443:1995
ASTM D3835-02
Specimen
Plastic melt forced through a capillary or slit die of known dimensions. A small representative sample is taken from the product to be tested.
The test specimen may be in any form that can be introduced into the bore of the cylinder such as powder, beads, pellets, strips of film, or molded slugs. In some cases it may be desirable to preform or pelletize a powder.
Apparatus
A heatable barrel, the bore of which is closed at the bottom end by an exchangeable capillary or slit die. The test pressure shall be exerted on the melt contained in this barrel by a piston, a screw or gas pressure
A capillary viscometer, the barrel, the capillary with a smooth straight bore, and the piston.
Test procedures
The polymer is introduced in the barrel preheated and forced through the capillary at a predetermined piston velocity. The pressure observed is registered with a pressure transducer. The pressure registered and the geometry of the capillary is used to calculate the shear stress (see equations in Table 5.9). The piston velocity and geometry of the barrel and capillary are used to calculate shear rate. For circular capillaries two corrections are performed, the RyderBagley and the WeissenbergRabinowitsch. For rectangular capillaries only the Weissenberg-Rabinowitsch correction is performed.
The polymer is introduced in the barrel preheated and forced to the capillary at a predetermined piston velocity. The pressure observed is registered with a pressure transducer. The pressure registered and the geometry of the capillary is used to calculate the shear stress.
Shear stress (Pa), Shear rate (s-1 ),Viscosity (Pa·s)
Shear stress, (Pa),Shear rate (s-1 ), Viscosity (Pa·s)
Values and Units
The piston velocity and geometry of the barrel and capillary are used to calculate the shear rate. For circular capillaries two correction are performed, the Ryder-Bagley and the Weissenberg-Rabinowitsch. For rectangular capillaries only the Weissenberg-Rabinowitsch correction is performed.
156
5 Melt Rheology
Table 5.11: Standard test methods for rubber-measurement of processing properties using capillary rheometry Standard
ASTM D5099-93(2003)
Scope
This test methods describes how capillary rheometry may be used to measure the rheological characteristics of rubber. Two methods are covered, Method A, which uses a piston type capillary viscometer, and Method B, which uses a screw extrusion type capillary viscometer. The two methods have important differences, as outlined by the test. The test methods cover the use of a capillary viscometer for the measurement of the flow properties of thermoplastic elastomers, unvulcanized rubber, and rubber compounds. These material properties are related to factory processing. Since the piston type capillary viscometers impart only a small amount of shearing energy to the sample, these measurements directly relate to the state of the compound at the time of sampling. Piston capillary viscometer measurements will usually differ from measurements with a screw extrusion type rheometer, which imparts shearing energy just before the rheological measurement. The capillary viscometer measurements for plastics are described in test method ASTM D3835.
Specimen
Massed specimen of raw or compounded unvulcsanized rubber for test method A -Piston extrusion capillary viscometer. Raw rubber or unvulcanized elastomeric compound formed into sheets on a two-roll mill for test method B - Screw extrusion capillary viscometer
Apparatus
A piston type capillary viscometer for test method A A screw extrusion capillary viscometer for test method B
Test procedures
Test method A - Piston extrusion capillary viscometer Unvulcanized rubber compound is placed in a temperature controlled cylinder fitted at one end with a conical transition section and a standard capillary die. The sample is driven through the die with the help of the piston while measuring or controlling the rate of extrusion and the pressure on the sample at the entrance of the die. Test method B - Screw extrusion capillary viscometer Unvulcanized rubber compound compound is formed into sheets on a two-roll mill. Strips cut from these sheets are fed to the extruder whose barrel is equipped with a temperature control. The end of the extruder is equipped with a transition conical section and a capillary die. A pressure transducer and temperature measuring device are placed in the chamber before the die. The rate of extrusion is calculated from the amount of extrudate collected over a timed interval. The rate of extrusion is controlled by adjusting the drive speed. Continued on next page
5.3 Rheometry
157
Standard
ASTM D5099-93(2003)
Values and Units
Viscosity curve (apparent and corrected) in log-log graph Corrected shear stress at 500 s-1 Corrected shear stress at 1000 s-1 Corrected viscosity at 500 s-1 , Corrected viscosity at 1000 s-1 Shear sensitivity, N Entrance effect, E
5.3.3 Rotational Rheometry A rotational rheometer is a particular type of rheometer in which the shear is produced by a drag flow between a moving part and a fixed one, including the following geometries: plateplate, cone-plate, and concentric cylinders. The main features of a rotational rheometer are the following: • The rotational rheometer can measure rheological properties under transient and steady state conditions. • It can reach the lowest shear rates of all rheometers. With this equipment it is possible to obtain shear rates typically from 10 -6 to 2.5 · 102 s-1 , so it is useful for macromolecular characterization (molecular weight distribution, relaxation spectra, and chain branching). • It can be used to measure the normal force in a polymer melt. • It offers a high degree of versatility because the following type of tests can be carried out: dynamic test (oscillation), flow test (rotation), static test, temperature sweep, torque sweep, frequency sweep, and time sweep. • It is used in conjunction with the capillary viscometer to obtain the complete viscosity curve. Rotational rheometers are used when more complex properties, such as normal stresses, are sought. There are two main types of rotational rheometers: the controlled rate rheometer (CRR), in which the strain or the shear strain is imposed and the stress is measured and the controlled stress rheometer (CSR), in which the stress is imposed and the strain or the shear rate is measured. Advances in rotational rheometer instrumentation have made it possible to have systems where both controlled rate and controlled stress can be programmed. Rotational rheometers include a temperature controller (which can be electrical heated plates, a Peltier system or an environmental test chamber), a test geometry (being the most common – plate-plate, cone-plate and concentric cylinders), a magnetic induction motor (being the most common a drag cup motor), an angular displacement measurement device (being the most typical an optical encoder), an electronic system to measure or control the torque, a mechanical frame, and a computer-based data acquisition and processing unit [17]. Parallel-plate rheometer: A parallel plate rheometer, schematically depicted in Fig. 5.27, is the geometrically simplest rotational rheometer, but mathematically it is more complex to analyze than its counterpart, the cone-and-plate rheometer.
158
5 Melt Rheology
Force
Torque
Ω θ
h R
Figure 5.27:
Schematic diagram of a parallel-plate rheometer
The plate-plate rheometer is sometimes the preferred system because of the following advantages: • Easy sample preparation of viscous materials and soft solids. • The shear rate can be easily changed by programming different rotational speeds or by adjusting the gap between plates or by changing the frequency. • Higher shear rates can be obtained before edge effects appear. • In conjunction with cone-plate geometry, the normal stress can be measured. • This geometry is preferred for viscous melts when small shear rates are required. The following equations are normally used in the flow mode: γ˙ R = M Ω
(5.37)
γR = M ϕ
(5.38)
where M is the geometric factor (R/h), h is the gap (mm), R is the external radius of the plate (mm), and ϕ is the deflection angle. Ω=
2π · N 60
(5.39)
where Ω is the angular speed (s -1 ) and N is the rotor speed (rpm). τ = Md · A · (
3+n ) 4
(5.40)
where Md is the torque (N· m), A is the geometric factor 2/R 3 (m-3 ), and n is the Power law exponent (Weissenberg correction). N1 -N2 = (
2Fn 1 dlnFn )(1+ ) πR2 2 dlnγ˙ R
(5.41)
where N1 -N2 is the second normal stress difference (Pa), and F n is the normal force (N).
5.3 Rheometry
159
Cone-plate geometry: The cone-plate rheometer is sometimes the preferred system because of the following advantages: • • • •
This rheometer allows the measurement of normal stresses. Homogeneous strain and simple equations. Useful for measurement of linear viscoelasticity, G(t, γ). Normally used for intermediate viscosity ranges. High-viscosity measurements are limited by the elastic problems in the borders and low-viscosity measurements are limited by inertial effects and sample loss in the borders. Torque
Force Ω
φ
θ
θo
Fixed plate Pressure transducers
R
Figure 5.28:
Schematic diagram of a cone-plate rheometer
The following equations are used in the flow mode: Ω β ϕ γR = β where Ω is the angular speed (s -1 ), β is the cone angle, and ϕ the deflection angle. γ˙ R =
3Md 2πR3 component) (Pa), and M d is the torque (N· m). τ=
where τ is the shear stress (τΦΘ
(5.42) (5.43)
(5.44)
3+n ) (5.45) 4 where Md is the torque (N· m), A is the geometric factor 2/R 3 , (m-3 ), and n is the Power law exponent (Weissenberg correction). τ = Md · A · (
2Fn ) (5.46) πR2 where N1 -N2 is the second normal stress difference (τ ΦΦ -τΘΘ component), (Pa), and F n is the normal force (N). N1 = (
160
5 Melt Rheology
Concentric cylinders geometry The concentric cylinders rheometer (also called the Couette rheometer) has the following characteristics: • • • •
Better for low-viscosity samples (under 100 (Pa·s)) Useful in high shear rates Gravity settling of a suspension has less effect than in a cone and plate rheometer Normal stress is difficult to measure Ω, T
Ri
L
Ro
Polymer
Figure 5.29:
Schematic diagram of a Couette rheometer.
The following equations are used in the flow mode: γ˙ = ϕ γ(R ˙ i) = γ(R ˙ i) =
Θ(Ro +Ri) 2(Ro − Ri)
2Ω Ri 2 (1 − ( R ) ) o 2Ω
n(1 −
2 Ri n (R ) ) o
n=
f or f or
dlnMd d(lnΩ)
(5.47) Ri > 0.99 Ro
0.50 <
Ri < 0.99 Ro
(5.48) (5.49) (5.50)
where Ω is the angular speed (s -1 ), Ro is the outside radius (mm), R i is the internal radius (mm), ϕ is the deflection angle, n is the Power law index (Weissenberg correction), and M d is the torque (N·m). τ=
Md 2πRi2 L
(5.51)
5.3 Rheometry
161
where τ is the shear stress (τΦΘ component) (Pa), M d is the torque (N·m), R i is the internal radius (mm), and L is the length of inside cylinder (mm). Temperature controllers: Because all rheological properties depend on temperature, the success of a rheological measurement starts with a very precise control of the temperature. Modern rheometers include the following types of temperature controls: • Fluid circulators – Fluid circulators use a thermostatic bath that precisely controls the temperature and circulates the fluid in the plates of the rheometer. Typically this control has a temperature range between -40 ◦ C and 250 ◦ C. • Peltier plate – This is the most common temperature control in rheometers and it uses the thermo-electric phenomenon called Peltier effect. The main limitation for a polymer analysis is the temperature range of -20 ◦ C to 200 ◦ C. The Peltier plate is a very accurate control, with a tolerance of about ±0.1 ◦ C, and has typical heating rates up to 20 ◦ C/min. • Electrical heated plates – In this temperature control, one plate is electrically heated. Typically, this control has a temperature range of -130 ◦ C up to 400 ◦ C. The low temperatures could be obtained with a special cooling device. • Thermal chamber – This is an oven that operates by convection and radiation. The main advantage of this system is the possibility of higher temperatures (typically from room temperature up to 1000 ◦ C, with the possibility of starting from -160 ◦ C using a special liquid (nitrogen) cooling device). Typical heating rates are up to 60 ◦ C/min.
0.5
-0.5
Figure 5.30:
Description of the dynamic test in a rotational rheometer
Operation modes: One of the advantages of the rotational rheometer is the versatility that allows different operation modes and tests. Rotational rheometry operation modes are presented in the following. Dynamic Test: These tests are done while subjecting one of the plates to oscillatory motion and by varying the frequency and the amplitude of the oscillation at isothermal conditions (see Fig. 5.30). This operation test can be used to obtain the complex viscosity curve, and its components, as a function of frequency. According to the Cox-Merz rule, in order to have a
162
5 Melt Rheology
complete viscosity curve, the complex viscosity curve can be superimposed with the viscosity curve obtained from a flow test and a capillary viscometer. This operation test obtains important viscoelastic functions, such as dynamic storage modulus G (ω) and dynamic loss modulus G (ω). Flow Test: In this test, one of the plates is subjected to a continuous rotational motion, enabling viscosity measurements as a function of the shear rate under realistic continuous flow. The main limitations are the centrifugal forces and border effects. Creep Test: The creep test, also called retardation, is a special test where the fluid is subjected to a stress step (stress controlled mode), and the strain variation during a period of time is registered. Eventually, the stress is released again and the strain recovery is registered. This experiment reflects viscoelastic behavior and the macromolecular characteristics of polymer melt, such as relaxation times. Relaxation Test: In the relaxation test, the fluid is subjected to a strain step (strain-controlled mode), and after some period of time the stress variation as a function of time, is registered. The relaxation test provides important information about the viscoelastic behavior and the macromolecular characteristics of polymer melts, such as stress overshoot. Torque Sweep: This test is useful in identifying the linear viscoelasticity region, which is the region where the compliance curves at different stresses can be super-imposed in a master curve independent of the applied stress. As a rule, the majority of rheological characterizations of a polymeric fluid are done in the linear viscoelastic region, so it is first recommended to determine the stress that limits this region. Frequency Sweep: This is a special dynamic test used to obtain the viscosity curve and important information about the viscoelastic behavior of a polymer melt. A sweep of frequency is normally done from very low to higher possible values without any border effect and under isothermal conditions. Temperature Sweep: In this test, the polymer sample is evaluated at a certain fixed frequency under a temperature program. The temperature sweep is particularly useful to determine some transition temperatures, such as glass transition and curing rate of thermosets and rubbers. The temperature sweep is also used to study curing of thermosets and degradation of polymers. Time Sweep: In this test, the polymer sample is evaluated at a given frequency during a period of time, and the change in the viscosity as a function of time is recorded. This particular test is useful in characterizing the thixotropic and rheopexic fluids. The thixotropic materials are fluids where the viscosity decreases with the time at a fixed shear rate, while rheopexic materials are fluids where the viscosity increases with time at a fixed shear rate. The standardized test to measure complex properties using parallel plate as well as coneand-plate rheometers are the ISO 6721 and ASTM D4440 tests presented in Table 5.12.
5.3 Rheometry
163
Table 5.12: Dynamic mechanical properties - complex shear viscosity using a parallelplate oscillatory rheometer Standard
ISO 6721-10:1999
ASTM D4440-01
Abstract/Scope
Determination of dynamic mechanical properties, such as complex shear viscosity as a function of frequency, strain amplitude, temperature, and time, using a parallel-plate oscillatory rheometer. This part specifies the general principles of a method for determining the dynamic rheological properties of polymer melts at angular frequencies typically in the range 0.01 10 Hz by means of an oscillatory rheometer with a parallel plate test geometry. Frequencies outside this range can be used if edge distortions and anomalies are not observed. The method is used to determine values of the dynamic rheological properties: complex shear viscosity h∗ , dynamic shear viscosity h’, the out-of-phase component of the complex shear viscosity h", complex shear modulus G∗ , shear loss modulus G" and shear storage modulus G".
Dynamic mechanical test to measure rheological properties of thermoplastic resins, such as complex viscosity, tan δ and significant viscoelastic characteristics as a function of frequency, strain amplitude, temperature, and time. This test method is valid for a wide range of frequencies, typically from 0.01 to 100 Hz.
This test method is intended for homogenous and heterogeneous molten polymeric systems and composite formulations containing chemical additives, including fillers, reinforcements, stabilizers, plasticizers, flame retardants, impact modifiers, processing aids, and other important chemical additives often incorporated into a polymeric system for specific functional properties, and which could affect the processability and functional performance. Apparent discrepancies may arise in results obtained under differing experimental conditions. Continued on next page
164
5 Melt Rheology
Standard
ISO 6721-10:1999
ASTM D4440-01
It is suitable for measuring complex shear viscosity values typically up to approximately 10 MPa.s.Test data obtained by this test method are relevant and appropriate for use in engineering design.
Without changing the observed data, reporting in full (as described in this test method) the conditions under which the data were obtained will enable apparent differences observed in another study to be reconciled. Test data obtained by this test method are relevant and appropriate for use in engineering design.
Specimen
In the form of a disc when produced by injection or compression molding or by cutting from sheet. Also, pellets or molten polymer.
A known amount of thermoplastic resin (molten powder or pellet, or solid preform disk) Molten polymer should be both homogeneous and representative.
Apparatus
Two concentric, rigid, circular parallel plates between which the specimen is placed. One of these plates oscillates at a constant angular frequency while the other remains at rest. An angular displacement and a torque measuring device record the strain and the stress during the test.
An apparatus to hold a molten polymer of known volume and dimensions so that the material acts as the elastic and dissipative element in a mechanically driven oscillatory system. The apparatus consists of the test fixtures (polished cone and plate, or parallel plates having either smooth, polished, or serrated surface), oscillatory deformation device, detectors (to determine stress, strain, frequency, and temperature), temperature controller and oven, Nitrogen, or other gas supply for purging purposes.
Test procedures
The specimen is held between the parallel plates and subjected to either a sinusoidal torque (controlled-stress mode) or sinusoidal angular displacement (controlledstrain mode).
Specimen is held between parallel plates or cone and plate and subjected to either a sinusoidal torque (controlled-stress mode) or sinusoidal angular displacement (controlledstrain mode). Continued on next page
5.3 Rheometry
Standard
Values and Units
5.3.4
165
ISO 6721-10:1999
ASTM D4440-01
In the controlled-stress mode the resultant displacement and the phase shift between torque and displacement are registered. In the controlledstrain mode the resultant torque and the phase shift between torque and displacement are registered. The equipment is able to measure important viscoelastic functions of the polymer melts.
In the controlled-stress mode the resultant displacement and the phase shift between torque and displacement are registered. In the controlled-strain mode the resultant torque and the phase shift between torque and displacement are registered. The equipment is able to measure important viscoelastic functions for the polymer melt under consideration.
Torque, angular displacement, angular frequency, shear stress, shear strain, shear storage modulus, shear loss modulus, complex shear modulus, dynamic shear viscosity, out-of-phase component of the complex shear viscosity, complex shear viscosity, and phase shift or loss angle all in SI units.
Dynamic moduli, complex viscosity, and tan δ as a function of the dynamic oscillation (frequency), percent strain, temperature, or time, all given in the standard SI units.
Extensional or Elongational Rheometry
It should be emphasized that the shear behavior of polymers measured with the equipment described in the previous sections cannot be used to deduce the extensional behavior of polymer melts. Extensional rheometry is the least understood field of rheology. Elongational or extensional properties are important when analyzing and understanding fiber spinning, thermoforming, film blowing, film casting, blow molding, and foaming. Several elongational rheometers have been designed to measure the elongational viscosity (or elongational stress) as a function of the elongational rate of deformation at different temperatures. However, the main challenges are how to obtain higher elongational rates, as well as simplifying the measurements, and making them reproducible. When measuring extensional viscosities, we can divide the techniques into direct and indirect methods. Direct measurement of elongational viscosity The following rheometers and elongational techniques, which fall under direct measuring techniques, have been proposed and used in the past [18]. • Extensional methods – These methods are more direct ways to measure the elongational behavior of polymers. Several types of extensional rheometers have been used, the following being the most common:
166
5 Melt Rheology
– Uniaxial extension – In this particular technique, the polymeric sample is stretched in one direction, and the elongational stress is measured under a defined elongation rate. Some examples of this type of rheometer are the Meissner rheometer [18], the vertical buoyancy bath [18], and the Sentmanat extensional rheometer (SER) [19, 20]. A schematic of Meissner’s extensional rheometer incorporating rotary clamps is shown in Fig. 5.31.
Spring εr = ln LA/LR Displacement sensor
Drive motor
LR LA
Sample Lo
Figure 5.31:
Schematic diagram of an extensional rheometer
– Lubricated compression – Another setup that can be used to measure extensional properties without clamping problems and without generating orientation during the measurement is the lubricating squeezing flow, which generates an equibiaxial deformation. To lubricate the material, the plates are usually coated with polydimethylsiloxane (silicone oil). A schematic of this apparatus is shown in Fig. 5.32.
Figure 5.32:
Schematic diagram of squeezing flow
– Biaxial and multiaxial extension – For high-viscosity polymers and rubber, it is possible to use a special rheometer with translating clamps that move in the orthogonal axis [18]. Rheometers have been built in the past that are able to move the sample in all directions, generating an equibiaxial extension of the sample.
5.3 Rheometry
167
– Bubble blowing – With this system, a sheet is clamped between two plates with circular holes and a pressure differential is introduced to deform it and blow the bubble into a test fluid. The pressure applied and deformation of the sheet are monitored over time and related to extensional properties of the material. The radius of the bubble allows the measurement of the extensional rate and the extensional strain, and the pressure difference and the interfacial tension allow one to determine the elongational stress [18]. The bubble blowing system is schematically depicted in Fig. 5.33. This test has been successfully used to measure extensional properties of polymer membranes for blow molding and thermoforming applications.
h
α
Figure 5.33:
R
Schematic diagram of sheet inflation
– Fiber spinning – This method is particularly useful for low-viscosity samples, where the polymer is continuously extruded and stretched by a rotating wheel. The diameter of the fiber as a function of the axial distance can be measured photographically and the force measured by the wheel with a load cell [18, 23, 24].
Figure 5.34:
Schematic diagram of a Rheotens extensional rheometer
168
5 Melt Rheology
– Rheotens – This technique was developed by the company G¨ottfert and consists of a tandem pulley system in which the melt that comes out from the circular capillary is pulled off between two sets of counter-rotating pulleys. A Rheotens rheometer is schematically depicted in Fig. 5.34. One of the pulleys is used to measure the torque [21]. The elongation viscosity as a function of the elongational rate can be determined with the help of software developed by Wagner and coworkers at the IKT, University of Stuttgart, Germany [22]. Indirect measurement of elongational viscosity Indirect methods for measuring elongational viscosity have been reported in the literature. The most popular method uses the pressure drop in sudden flow contraction and the stagnation flow. • Flow stagnation – This technique uses the principle that steady extensional deformations can be created by impinging two liquid streams, such as depicted in Fig. 5.35 [18]. Although with stagnation flows it is only possible to measure steady extensional viscosity, there is great interest in this technique because high elongational rates and low-viscosity samples can be studied.
Stagnant region
Figure 5.35:
Schematic diagram of an impinging flow
• Entrance flows – This method is based on the pressure losses in sudden flow contractions and can be considered as a special case of a flow stagnation technique. The most common methods are the following: – Cogswell’s method – According to the theory developed by Cogswell, the elongational viscosity is obtained from the pressure drop in a sudden flow contraction, such as a capillary with an inlet angle of 90 ◦ , as schematically depicted in Fig. 5.36 [18, 23, 24, 25]. According to Cogswell, the elongational rate and the elongational viscosity can be estimated by using the following equations:
5.3 Rheometry
169
Pressure transducer Recirculation or vortex zone Entrance pressure drop
Figure 5.36:
Schematic diagram of the Cogswell method
˙ =
2 4η γ˙ ap 3(n + 1)ΔPe
9(n + 1)2 ηe = 32η γ˙ ap =
4Q πR3
ΔPe γ˙ ap
(5.52) (5.53)
(5.54)
where ˙ is the elongational rate, η e is the elongational viscosity, η is the viscosity, γ˙ ap is the power law index, ΔP e is the pressure drop at capillary inlet (obtained from Bagley correction extrapolation at L/D = 0) , and n is the apparent shear rate – Binding’s method – This method is based in the theory developed by Cogswell, but it is a more accurate method since Binding does not neglect the WeissenbergRabinowitsch correction. However, this leads to more complex calculations [26, 27]. To model the shear viscosity, and the elongational viscosity Binding arbitrarily assumes a Power law model. – Semihyperbolically converging die – In this technique, schematically depicted in Fig. 5.37, the polymer flows through a cylindrical, converging die whose semihyperbolic shape leads to a shear-free flow within the die, assuming wall slip conditions. From the analysis of the relevant flow equations in the die, the use of a numerical method (typically the finite element method (FEM)) and the use of a constitutive equation, the elongational viscosity can be measured [28].
170
5 Melt Rheology
Figure 5.37:
Schematic diagram of a semihyperbolically converging extensional rheometer
INDUSTRIAL APPLICATION 5.2
Molecular Weight Distribution Comparison Using Rheometry Characterization Here, we will perform an analysis of molecular weight distributions of three different polypropylenes using storage modulus, G , and loss modulus, G , measurements as a function of the frequency. These measurements are presented in Table 5.13. Table 5.13:
Dynamic modulus for three different polypropylenes
PP - A G (Pa)
G" (Pa)
Frequency (rad/s)
0.1000
585
1340
0.2150
1190
2130
Frequency (rad/s)
PP - B G (Pa)
G" (Pa)
Frequency (rad/s)
PP - C G (Pa)
0.0300
585
1340
0.0300
1170
2680
0.0646
1190
2130
0.0646
2380
4250
G’" (Pa)
0.4640
2200
3330
0.1390
2200
3330
0.1390
4410
6650
1
3900
5010
0.3000
3900
5010
0.3000
7790
10000
2.1500
6480
7340
0.6460
6480
7340
0.6460
13000
14700
4.6400
10200
10300
1.3900
10200
10300
1.3900
20500
20500
10
15600
14200
3
15600
14200
3
31100
28400
21.5000
23000
19400
6.4600
23000
19400
6.4600
45900
38700
46.4000 100
32900 45000
25900 33700
13.9000 30
32900 45000
25900 33700
13.9000 30
65700 90100
51700 67300
5.3 Rheometry
171
Using the crossover 5 of storage modulus G and loss modulus G , the molecular weight distribution (MWD) of the different polypropylenes was compared. Figure 5.38 illustrates the relation between the crossover of G and G and the molecular weight distribution. As shown in the figure, as the molecular weight increases, the crossover modulus, G c , shifts to a lower frequency; and as MWD narrows, the crossover modulus shifts to a higher value.
Figure 5.38:
Correlation between the molecular weight distribution and the crossover modulus
Figure 5.39 presents plots of the dynamic modulus presented in Table 5.13. The crossover modulus allows the comparison of the molecular weight distribution (MWD) of the three polypropylenes.
Figure 5.39:
Comparison of the crossover modulus for the three different polypropylenes
Conclusions: When comparing the crossover modulus for the three polypropylenes, the following conclusions could be obtained: 5 The
crossover point is where G /G" = 1.
172
5 Melt Rheology
– The molecular weight for the three polypropylenes is M W C > M W B > M W A, because the order of the crossover on the frequency axis was G cA > GcB > GcC . – The molecular weight distribution of polypropylene C was narrower than polypropylenes A and B, because the order of the crossover on the modulus axis was G cC > GcA = GcB . – Because polypropylene C exhibited the higher molecular weight and the narrower MWD, it will more easily result in flow instabilities and can exhibit higher elastic effects in the molten state. – The shear viscosity curve can also be used to interpret molecular weight distributions. The smaller Newtonian plateau and the more gradual pseudoplastic decrease of viscosity exhibited by a polymer reflect a wider molecular weight distribution. The higher zero viscosity (the viscosity value in the Newtonian plateau) exhibited by the polymer reflects a higher average molecular weight of the polymer melt. To illustrate the correlation, Fig. 5.40 compares the viscosity curves for two different polymers at the same temperature. When looking at the figure, one can deduce that polymer B has the wider MWD because of its narrow Newtonian plateau and its more gradual pseudoplastic decrease of viscosity.
Figure 5.40: Correlation between MWD and the viscosity curves for two polypropylene melts at the same temperature
INDUSTRIAL APPLICATION 5.3
Flow Instabilities Study in a Thermoplastic Polymer In this case study, a mass flow of 70 kg/h of polypropylene was extruded through a rectangular die of 50 cm width, 10 cm length and 0.2 cm height, at 220 ◦ C. The particular PP to be extruded had the following rheological information based on a Bird-Carreau-Yasuda model: A = 4254 (Pa· s), B = 0.22 (s) , C = 0.63 , Tref = 243 (◦ C) , and U = 46015 (J/mol). The density of polypropylene at 220 ◦ C is 0.75 g/cm3 . The estimated pressure drop and the prediction of flow instabilities at the exit of the die were required under the specified conditions.
5.3 Rheometry
173
The volumetric flow was calculated as follows V =
m ˙ 70 · kg · h · 1000 · g · cm3 cm3 = = 25.93 . ρ h · 3600 · s · kg · 0.75 · g s
(5.55)
The shear rate for a non-Newtonian polymer was calculated with the following approximated equation: 6·V 0.772 · 6 · 25.93 · cm3 = = 60.04 · s-1 . 2 W ·h s · 50 · cm · 0.22 · cm2 The shear viscosity can be calculated using γ˙ =
1 U 1 ( − ) aT = exp R T Tref 1 1 46015 · J · mol · K ( − ) = exp mol · 8.3141 · J (220+273.15)K (243+273.15)K
(5.56)
(5.57)
= 1.65 η=
A · aT 4254Pa · s · 1.65 = = 979 · Pa · s. 0.63 (1+B · aT γ) ˙ c (1+0.22 · s · 1.65 · 64.04 s )
(5.58)
The pressure drop can be estimated using
ΔP =
12 · V · η · L 12 · 25.93 · cm3 · 979 · Pa · s · 10 · cm = = 7614111 · Pa W · h3 s · 50 · cm · (0.2)3 · cm3
7614111 · Pa · bar = 76.14 · bar. 105 Pa The shear stress can now be computed using ΔP =
(5.59)
(5.60)
MPa 60.04 · 6 = 0.06 · MPa. (5.61) s 10 · Pa Conclusions: An estimated pressure drop of 76.14 bar was predicted for the particular die and the given extrusion conditions. Because the calculated shear stress was below the critical shear stress for flow instabilities of polypropylene (0.13 MPa, see Table 5.1), flow instabilities were not predicted under the conditions stated in this case. τ = η · γ˙ = 979 · Pa · s
INDUSTRIAL APPLICATION 5.4
Modeling the Shear Viscosity Curves of a Polypropylene Here, a Bird-Carreau-Yasuda model was used to fit the data for a polypropylene polymer melt.
174
5 Melt Rheology
Table 5.14:
Capillary rheometry data for a polypropylene
Temperature, (◦ C) shear rate, (s-1 )
200 Viscosity, (Pa· s)
210 Viscosity,(Pa· s)
220 Viscosity, (Pa· s)
10
591
501
429 420
20
575
490
50
532
458
396
70
508
440
382
100
475
415
363
200
395
351
313
500
270
248
227
700
227
210
194
1000
185
172
161
2000
120
113
106
5000
64
61
58
7000
50
48
46
10000
39
37
35
The curve was obtained using a capillary viscometer, and the data was corrected according to the Ryder-Bagley and Weissenberg-Rabinowitsch correction. The data are presented in Table 5.14.
– The parameters of the Bird-Carreau-Yasuda model were obtained by fitting the experimental viscosity values and the calculated viscosities. Since polypropylene is a semicrystalline polymer, the best model to correlate the temperature dependence of viscosity is the Arrhenius model 1 U 1 aT = exp ( − ) (5.62) R T Tref η=
A · aT (1+B · aT γ) ˙ c
(5.63)
– The values of A, B, and C were obtained by minimization of the error between experimental data and the calculated data for the different shear rates and temperatures, according to the following error metrics Error =
M
(η calc − η exptal )
(5.64)
i=1
– When fitting the parameters of the Bird-Carreau-Yasuda model with the experimental data, a non-linear optimization was done using a commercial equation solving software.
5.3 Rheometry
175
Because of the non-linearity of the model, the initial guess values were very important for the convergence of the optimization. It was necessary to impose some limits to the A, B, and C values, as follows: • A is greater than zero • B is greater than zero • C is within the range of 0 to 1 (typically between 0.3 and 0.7) It was also important to limit the activation energy, U , to be greater than zero. Table 5.15 presents the optimized parameters of the Bird-Carreau-Yasuda model for the given polypropylene data. Figure 5.41 shows a comparison between the experimental data and the model.
Figure 5.41: Experimental data (symbols) and the model prediction (lines) for the viscosity of a polypropylene polymer melt
Table 5.15:
Parameters of the Bird-Carreau-Yasuda model for a polypropylene Parameter
Value
A, (Pa · s)
608.758667
B,(s) C
0.003946 0.744911
U , (J/mol)
32052.3943
Tref , (◦ C)
200
Conclusions: The obtained parameters of the Bird-Carreau-Yasuda model reproduced with good precision the experimental viscosity data for all three temperatures.
176
5 Melt Rheology
INDUSTRIAL APPLICATION 5.5
Regression Analysis of Rheological Data to Obtain the Phan-Thien and Tanner Multimode Model Parameters The parameters of the Phan-Thien and Tanner multimode model for a particular polypropylene to be used for fiber spinning applications were obtained by fitting the following rheological data: – Material – Homopolymer polypropylene with a MFI 16 g/10 m at 230 ◦ C/2.16 kg – Shear viscosity and Cogswell’s elongational viscosity – These data were measured in a capillary viscometer using a 1 mm circular capillary with 10, 20, 30, and 40 mm lengths. – Elongational viscosity – Measured at low elongational rates (in the Newtonian range, below 0.1 s-1 ) and computed using the Trouton viscosity equation. This relationship states that the elongational viscosity at very low elongational rates is equal to 3 times the shear viscosity. – Shear viscosity and complex modulus – These data was measured in a rotational rheometer with a 25 mm diameter plate-plate, 1 mm gap, 1% deformation (in the linear viscoelasticity range), and frequency within the range from 0.005 and 100 rad/s. The rheological data are presented in Fig. 5.42. The unfilled circles in the figure correspond to the shear viscosity obtained by capillary rheometry, and the filled circles correspond to the shear viscosity obtained by rotational rheometry and by applying the Cox-Merz principle of complex viscosity. The unfilled triangles in the figure correspond to loss modulus G obtained by rotational rheometry; the unfilled squares correspond to storage modulus G obtained by rotational rheometry; the unfilled rhomboids correspond to the elongational viscosity estimated by the Cogswell’s method; and the filled rhomboids correspond to the elongational viscosity estimated with the Trouton relationship. The parameters of the model were obtained by fitting the rheological data, according to the following procedure: – The values of λ i and Gi were obtained by fitting the storage and loss modulus, according to the equations
G =
N Gi (λi ω)2 i=1
1+(λi ω)2
N Gi λi ω G = 2 1+(λ i ω) i=1
(5.65)
5.3 Rheometry
Figure 5.42:
177
Dynamic test in a rotational rheometer [23, 24]
and the restriction ηo =
N
(5.66)
Gi λi .
i=1
– The values of ξ i were obtained by fitting shear viscosity according to η=
N i=1
Gi λi . 1+ξi (2 − ξi )(λi ω)2
(5.67)
– The λi , Gi and ξi values were obtained by minimization of the error between experimental data and the calculated data using the previous equations for the different frequencies, using the error metrics
Error =
M
Gicalc
2
M
−1 + Giexptal i=1 2 M ∗calc η + −1 . ∗exptal η i=1 i=1
2
Gicalc
Gi exptal
−1 (5.68)
– The values of i were obtained by fitting the Trouton viscosity at low elongational rates and the Cogswell’s elongational viscosity. To fit the elongational viscosity, it was necessary to solve by iteration the non-linear equation resulting from applying the Phan-Thien and Tanner equations to an elongational, uniaxial, and uniform flow at steady state:
178
5 Melt Rheology
i exp (τzzi +2τrri ) τzzi − 2λi (1 − ξi )τzzi ˙ = 2λi Gi , ˙ Gi
i (τzzi +2τrri ) τrri − 2λi (1 − ξi )τrri ˙ = 2λi Gi , ˙ exp Gi τzz =
N
τzzi
τrr =
i=1
N
η=
τrri
i=1
τzz − τrr .
˙
(5.69) (5.70)
(5.71)
– The ξi values were obtained by minimization of the error between experimental data and the calculated data using the above equations for the different frequencies, according to the error metrics Error =
M ηecalc i=1
Table 5.16: 230 ◦ C)
2
ηeexptal
−1
(5.72)
.
Parameters for Phan-Thien and Tanner model (PP PROPILCO 18H86 at
i
λi
i
ξi
Gi
1
2.51E-04
1.00E+00
8.60E-02
1.22E+05
2
1.66E-03
1.00E+00
3.06E-01
4.02E+04
3
5.94E-03
8.00E-01
9.43E-01
1.53E+04
4
2.07E-02
2.00E-01
5.89E-01
6.39E+03
5
9.00E-02
3.50E-02
8.30E-02
2.88E+03
6
4.86E-01
3.00E-02
2.62E-01
3.62E+02
7
2.71E+00
3.00E-02
9.99E-01
3.44E+01
When fitting the parameters of the Phan-Thien and Tanner multimode model with the experimental data, a non-linear optimization was done using a commercial equationsolving software. Table 5.16 shows the parameters of the Phan-Thien and Tanner multimode model for the given polypropylene. The solid lines of Fig. 5.42 correspond to the Phan-Thien and Tanner multimode model predictions. Conclusions: The Phan-Thien and Tanner multimode model agreed very well with the given experimental data. Small oscillations are visible because of the discretization of the relaxation spectra.
5.3 Rheometry
179
INDUSTRIAL APPLICATION 5.6
Modeling the Shear Viscosity Curves and Their Application in Injection Molding In this case study, the cold runners and gates of a six-cavity mold presented in Fig. 5.43 needed to be rheologically balanced, such that every cavity filled at the same time. The polymer melt data is given by, – Material – Injection molding grade polyamide 6 – Rheological data – The rheological properties were modeled at the injection temperature using the Bird-Carreau-Yasuda model with parameters, A = 373 (Pa· s), B = 0.12 s and C = 0.35. The details of the regression procedure to obtain the Bird-CarreauYasuda model parameters were presented in in a case study above. – Processing conditions and properties – The following injection molding conditions were set: part weight 20 g, total length of channel 250 mm, density at melt temperature 1.1 g/cm3 , injection molding speed 100 mm/s and thickness 2.5 mm. – Geometry of gates and runners – The dimensions of the runners and gate system are presented in Table 5.17. The diameters of channels 1 and 3 were used for balancing the runners and gate system. The values presented are the final values obtained after the iteration process. Table 5.17:
Dimensions of the runners and gate system of the six-cavity mold Diameter (mm)
Length (mm)
Number of channels
Channel 1
5.11
100
2
Channel 2
7
150
2
Channel 3
4.88
100
4
Gate I
1
1
2
Gate II
2
1
4
The first step is to calculate the total volume of the part, runners, and gates of the given mold system. The total volume was calculated by adding the volume of the six parts (calculated with the weight of the part and the melt density), the volume of the different channels and volume of the different types of runners (calculated with the given geometry),
VT ot =
N Wpart
i
i=1
ρmelt
− 1+
M j=1
Vchannelj +
P k=1
Vgatesk .
(5.73)
180
5 Melt Rheology
Figure 5.43:
Six-cavity mold to be rheologically balanced
The injection time was calculated by dividing the injection molding speed by the channel length, Vinj . (5.74) L The volumetric flow was calculated by dividing total volume by the injection time, t=
VT ot . V˙ = t Table 5.18: mold
(5.75)
Calculations for balancing the runners and gate system of the six-cavity
Diameter (mm)
Length (mm)
# of Channels
Flow rate (cm3 /s)
Shear Rate (1/s)
Viscosity (Pa· s)
Pressure drop (Bar)
Channel 1
5.11
100
2
8.8
547.4
85.8
45.1
Channel 2
7
150
2
17.6
426.7
93.4
41.9
Channel 3
4.88
100
4
8.8
629.3
81.7
51.7
Gate I
1
1
2
8.8
73186.6
15.5
55.8
Gate II
2
1
4
8.8
9148.3
32.2
7.2
Total
52.8
The volumetric flow of channels 1 and 3 was calculated as the total volume divided by 6. The volumetric flow of channels 2 was calculated as the total volume divided by 3. The shear rate for each channel and gate was approximated using 4 · V˙ . (5.76) π · R3 The viscosity was calculated with the Bird-Carreau-Yasuda model, and the pressure drop for each channel and gate was obtained with the expression γ˙ = 0.815 ·
η= ΔP =
A , (1+B · γ) ˙ c 8 · V˙ · η · L . π · R4
(5.77) (5.78)
5.3 Rheometry
181
Using an iterative scheme with the diameters of channels 1 and 3 and with the diameter of channel 2 fixed, the runners and gate system were obtained. Because the runners and gate system were balanced, the pressure drop through every channel was the same and equal to 100.9 bar. The maximum recommended shear rates for gates are presented in Table 5.19. For a PA6 a maximum shear rate of 60000 1/s is recommended. According to the calculations, the higher shear rate 73186.6 1/s was obtained. Table 5.19:
Maximum shear stress and shear rates for various polymers [29, 30] Polymer
Max. shear stress, (Pa)
Max. shear rate, (1/s)
PP
250,000
100,000
HDPE
200,000
40,000
LDPE
100,000
40,000
Flexible PVC
150,000
20,000
Rigid PVC
200,000
20,000
PS
250,000
40,000
HIPS
300,000
40,000
SAN
300,000
40,000
ABS
300,000
50,000
PA6
500,000
60,000
PA66
500,000
60,000
PET
500,000
6,000
PBT
400,000
50,000
PC
500,000
40,000
PMMA
400,000
40,000
PPS
345,000
50,000
PSU
500,000
50,000
PUR
250,000
40,000
Conclusions: Although the cold runners and gates system of the six-cavity mold was balanced, a shear rate at gate 1 exceeded the maximum recommended value; hence, a redesign of this gate must be done. INDUSTRIAL APPLICATION 5.7
Dispersion of a Polymer Blend Using a Single-Screw Extruder In this case study a 50% PP and 50% LDPE polymer blend was supposed to be processed in a 45-mm single-screw extruder. It was necessary to predict beforehand which polymer could be used as the dispersed phase to guarantee the required product
182
5 Melt Rheology
quality. The intended operating conditions of the extruder and the barrier gap geometry of the screw in the metering zone were: Temperature of the melt is 210 ◦ C Screw rotational speed, N = 45 rpm = 0.8 rev/s Barrier gap, δ = 0.3 mm The first step is to estimate the viscosities of both polymeric materials at the extrusion shear rate (γ) ˙ within by the 45-mm screw. The shear rate was calculated by the expression γ˙ =
π·D·N π · 45 mm · 0.8rev/s = = 377s-1 . δ 0.3 mm
(5.79)
Figures 5.44 and 5.45 present the viscosity curves of LDPE and PP as a function of shear rate and temperature of the melt.
Figure 5.44:
Viscosity curve of LDPE
The viscosity values of the polymer melts shown in Eq. 5.80 can be obtained from the curves in Fig. 5.44 and 5.45.
Figure 5.45:
Viscosity curve of PP
5.3 References
Figure 5.46:
183
Grace diagram [1]
ηLDP E = 80(Pa · s) ηP P = 250(Pa · s)
(5.80)
The viscosity ratios were also calculated to predict the dispersed phase reading in the Grace diagram and are shown in Eq. 5.81. ηLDP E 80 = 0.32 = ηP P 250 ηP P 250 = 3.12 = ηLDP E 80
(5.81)
From the Grace diagram in Fig. 5.46, it was clear that PP cannot be employed in the dispersed phase because the viscosity ratio η P P over ηLDP E was near the 3.8 limit. Because the ratio ηLDP E over ηP P was the lowest, for LDPE it is likely to be selected for the dispersed phase. Conclusion: The predicted dispersed phase for the physical blend of 50% PP and 50% LDPE at the given operating conditions was LDPE, and the matrix or continuous phase was PP.
References 1. F.T. Trouton. Proc. Roy. Soc. A, 77, 1906. 2. H. Muenstedt. Rheol. Acta, 14:1077, 1975. 3. C.D. Han and K.W. Len. J. Appl. Polym. Sci., 29:1879, 1984. 4. J.M. Castro and C.W. Macosko. AIChE J., 28:250, 1982. 5. J.M. Castro, S.J. Perry, and C.W. Macosko. Polym. Comm., 25:82, 1984. 6. G.K. Batchelor. Annu. Rev. Fluid Mech., 6:227, 1974.
184
5 Melt Rheology
7. A. Einstein. Ann. Physik, 19:549, 1906. 8. P. Geisb¨usch. PhD thesis, IKV-RWTH-Aachen, Germany, 1980. 9. E. Guth and R. Simha. Kolloid-Zeitschrift, 74:266, 1936. 10. E. Guth. Phys. Rev., 53:321, 1938. 11. R.K. Gupta. Flow and rheology in polymer composites manufacturing. Elsevier, Amsterdam, 1994. 12. W.J. Milliken and R.L. Powell. Flow and rheology in polymer composites manufacturing. Elsevier, Amsterdam, 1994. 13. J.F. Agassant, P. Avenas, J.-Ph. Sergent, and P.J. Carreau. Polymer Processing - Principles and Modeling. Hanser Publishers, Munich, 1991. 14. J. P. Sergent J. F. Agassant, P. Avenas and P. J. Carreau. Polymer processing: principles and modelling. Hanser Publishers, Munich, 1991. 15. R. I. Tanner, N. Phan-Thien. J. Now-Newt Fluid Mech, 2:353, 1977. 16. D. Acierno, F. P. La Mantia, G. Marrucci, and G. Titomanlio. J. Now-Newt Fluid Mech, 1:125 –147, 1977. 17. H. A. Barnes and D. Bell. Controlled-stress rotational rheometry: A historical review. KoreaAustralia Rheol. J, 15(4):285–336, 2003. 18. C.W. Macosko. RIM Fundamentals of reaction injection molding. Hanser Publishers, Munich, 1989. 19. B. N. Wang, M. Sentmanat. and G. H. McKinley. Measuring the transient extensional rheology of a LDPE melt using the SER universal testing platform. J. Rheol, pages 1–29, 2004. 20. M. Sentmanat. Miniature universal testing platform: from extensional melt rheology to solid-state deformation behavior. Rheol Acta, 43:657–669, 2004. 21. RHEOTENS 71.97. The New Tensile Tester for Polymer Melts. G¨ottfert, Germany. 22. A. Bernnat. Polymer Melt Rheology and the Rheotens Test. PhD thesis, Institut f¨ur Kunststoffverarbeitung, 2001. 23. J. Sierra. Development of biconstituent fibers of polypropylene and polyamide 6 for high speed fiber spinning. PhD thesis, University of the Vasc Country, Spain, 2005. 24. M. P. Noriega, J. D. Sierra, I. D. L´opez, and I. Katime. Biconstituent fibers from polypropylene and polyamide 6: fiber spinning modeling and properties. WAK Kunststoffetechnik, 2:1–28, 2006. 25. F. N. Cogswell. Polymer melt rheology: A guide for industrial practice. John Wiley & Sons, 1981. 26. D. Binding. An approximate analysis for contraction and converging flows. J. Non-Newtonian Fluid Mech, 27:173–189, 1988. 27. M. Gupta. Effect of elongational viscosity on axisymmetric entrance flow of polymers. Polym. Eng. Sci., 40, 23, 2000. 28. K. Feigl, F.X. Tanner, B.J. Edwards, and J.R. Collier. A numerical study of the measurement of elongational viscosity of polymeric fluids in a semihyperbolically converging die. To appear in Journal of Non-Newtonian Fluid Mechanics. 29. H. Rees. Mold Engineering. 2nd edition Hanser Publishers, Munich, 2002. 30. J. P. Beaumont. Runner and Gating Design Handbook: Tools for Successful Injection Molding. 2nd edition Hanser Publishers, Munich, 2007. 31. T. A. Osswald and G. Menges. Material Science of Polymers for Engineers. Hanser Publishers, Munich, 2nd edition, 2003.
CHAPTER 5
MELT RHEOLOGY
Rheology is the field of science that studies fluid behavior during flow-induced deformation. From the variety of materials that rheologists study, polymers have been found to be the most interesting and complex. Polymer melts are shear thinning, viscoelastic, and their flow properties are temperature dependent. Viscosity is the most widely used material parameter when determining the behavior of polymers during processing. Since the majority of polymer processes are shear-rate dominated, the viscosity of the melt is commonly measured using shear deformation measurement devices. However, there are polymer processes, such as blow molding, thermoforming, and fiber spinning, that are dominated by either elongational deformation or by a combination of shear and elongational deformation. In addition, some polymer melts exhibit significant elastic effects during deformation. Rheometry is the aspect of rheology that measures by means of analytical techniques the rheological properties required for process layout and assessment. Furthermore, rheological characterization is an important tool for studying the molecular weight distribution of polymers, as well as for troubleshooting, optimizing, and designing processing equipment. Some industrial applications and applications are discussed later in this chapter.
128
5 Melt Rheology
5.1 BASIC CONCEPTS AND TERMINOLOGY This section summarizes the basic concepts and terminology commonly used in the field of rheology. Stress: Stress, τ , is the amount of force applied in a particular area. Depending of the type of flow, it is common to define the following stresses: shear stress and normal stress. Strain: Strain, γ, quantifies the deformation imposed on a material element. Depending of the type of flow, we distinguish between shear strains and elongational strains. Strain rate: The strain strain rate, γ, ˙ is sometimes also referred to as a rate of deformation: or shear rate. The natural response of a solid is a finite strain; however, for a liquid, the natural response is a strain rate rather than a finite strain. Viscosity: Viscosity, η, is the rheological property that relates the stress to the strain rate. In a Newtonian fluid, the deviatoric stresses that occur during deformation, τ , are directly proportional to the rate of deformation tensor, γ, ˙ τ = μγ. ˙
(5.1)
For Newtonian liquids, the viscosity, μ, is considered to be only dependent on temperature. However, the viscosity of most polymer melts is shear thinning in addition to being temperature dependent. The shear thinning effect is the reduction in viscosity at high rates of deformation. This phenomenon occurs because, at high rates of deformation, the molecules are stretched out and disentangled, enabling them to slide past each other with more ease, hence lowering the bulk viscosity of the melt. To take into consideration these non-Newtonian effects, it is common to use a viscosity which is a function of the strain rate and temperature to calculate the stress tensor in Eq. 5.1 τ = η(γ, ˙ T )γ, ˙
(5.2)
where η is the non-Newtonian viscosity and γ˙ the magnitude of strain rate or rate of deformation tensor defined by 1 γ˙ = II, (5.3) 2 where II is the second invariant of the strain rate tensor defined by γ˙ ij γ˙ ji . (5.4) II = i
j
The strain rate tensor components in Eq. 5.4 are defined by γ˙ ij =
∂ui ∂uj + . ∂xj ∂xi
(5.5)
The temperature dependence of the polymer’s viscosity is normally factored out as η(γ, ˙ T ) = f (T )η(γ), ˙
(5.6)
5.1 Basic Concepts and Terminology
Figure 5.1:
129
Viscosity curve for a polypropylene
100004 10 Compression molding Calendering
Injection molding
Extrusion
Fiber spinning
220 oC ABS
10003 10
260 oC 300 oC PC 340 oC 260 oC PA6
1002 10 300 oC
101 10
10
100
1000
10000
100000
Shear rate γ˙
Figure 5.2:
Viscosity curves for a selected number of thermoplastics
where for small variations in temperature, f (T ) can be approximated using an exponential function such as f (T ) = e-a(T -T0 ) .
(5.7)
Figure 5.1 illustrates the typical viscosity curve of a polymeric melt at three different temperatures. As presented in this figure, the viscosity curve is best presented in double-log graphs. Figures 5.2 through 5.6 show the shear thinning behavior and temperature dependence of the viscosity for a selected number of thermoplastics. Elongational viscosity: In polymer processes such as fiber spinning, blow molding, thermoforming, foaming, certain extrusion die flows, and compression molding with specific processing conditions, the major mode of deformation is elongational. To illustrate elongational flows, consider the fiber-spinning process shown in Fig. 5.7.
130
5 Melt Rheology
100000 PE-HD Low MFI (230 oC)
Pa•s
10000
1000 PP Low MFI (230 oC) PP High MFI (230 oC)
100
PE-HD High MFI (230 oC)
10 1
10
100
1000
10000
s -1
100000
Rate of deformation
Figure 5.3:
Viscosity curves for PE-HD and PP with low and high MFI at 230 ◦ C
10000 Pa•s
SAN (250 oC) ABS (250 oC)
PBT (250 oC)
1000
o
PS (250 C)
PA6 (250 oC) 100
10
1 1
10
100
1000
10000
s -1
Rate of deformation
Figure 5.4:
Viscosity curves for a selected number of thermoplastics at 250 ◦ C
100000
5.1 Basic Concepts and Terminology
131
10000 ASA (240 oC) Pa•s PES (340 oC)
1000
SB (240 oC)
PA66 (290 oC)
100
10
1 1
10
100
1000
s -1
10000
100000
Rate of deformation
Figure 5.5:
Viscosity curves for selected thermoplastics
100000
PMMA Low MFI (200 oC)
Pa•s 10000
PMMA High MFI (200 oC) 1000
PMMA High MFI (220 oC)
PAEK (400 oC) PMMA Low MFI (240 oC)
100
LCP (340 oC) 10
1 1
10
100
1000
10000
s -1
Rate of deformation
Figure 5.6:
Viscosity curves for PMMA, PAEK, and LCP
F
Figure 5.7:
Schematic diagram of a fiber-spinning process
100000
132
5 Melt Rheology
A simple elongational flow is developed as the filament is stretched with the following components of the rate of deformation: γ˙ 11 = - ˙ γ˙ 22 = - ˙
(5.8)
γ˙ 33 = 2 ˙ where ˙ is the elongation rate, and the off-diagonal terms of γ˙ ij are all zero. The diagonal terms of the total stress tensor can be written as σ11 = -p-η ˙ σ22 = -p-η ˙ σ33 = -p+2η ˙
(5.9)
Since the only outside forces acting on the fiber are in the axial or 3-direction, for the Newtonian case, σ11 and σ22 must be zero. Hence,
and
p = -η ˙
(5.10)
σ33 = 3η ˙ = η¯ , ˙
(5.11)
which is known as elongational viscosity or Trouton viscosity [1]. This is analogous to elasticity where the following relation between elastic modulus, E, and shear modulus, G, can be written E = 2(1+ν), (5.12) G where ν is Poisson’s ratio. For the incompressibility case, where ν = 0.5, Eq. 5.12 reduces to E = 3. (5.13) G 5•108
Viscosity (Pa-s)
108
Elongational test
μo = 1.6•108 Pa-s ηo = 5.5•107 Pa-s
5•107 ηo = 5•10 7 Pa-s
Shear test
107 5•106
106 102
Figure 5.8:
μo = 1.7•108 Pa-s
T = 140 °C Polystyrene I Polystyrene II 5•102 103
104 Stress, τ, σ (Pa)
105
5•105
Shear and elongational viscosity curves for two types of polystyrene
5.1 Basic Concepts and Terminology
133
6 LDPE
Log(Viscosity) (Pa-s)
5 Ethylene-propylene copolymer
4
PMMA POM
3
PA66
2 3
4
5
6
Tensile stress , log σ (Pa)
Figure 5.9:
Elongational viscosity curves as a function of tensile stress for several thermoplastics 0.8 0.7
60 °C
Degree of cure, c
0.6 50 °C
0.5 0.4
40 °C 0.3 0.2 0.1 0.0 10 -1
Figure 5.10: temperatures
10 0 10 1 Cure time (min)
10 2
Degree of cure as a function of time for a vinyl ester at various isothermal cure
Figure 5.8 [2] shows shear and elongational viscosities for two types of polystyrene. In the region of the Newtonian plateau, the limit of 3, shown in Eq. 5.11, can be seen. Figure 5.9 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 difficult task. For example, to maintain a constant strain rate, the specimen must be deformed both uniformly and exponentially. In addition, a molten polymer must be tested completely submerged in a heated, neutrally buoyant liquid at a constant temperature.
134
5 Melt Rheology
10 60 °C 50 °C
Viscosity (Pa-s)
40 °C
100
10 -1 0
Figure 5.11: temperatures
0.1
0.2 0.3 0.4 0.6 Degree of cure, c
0.5
0.7
Viscosity as a function of degree of cure for a vinyl ester at various isothermal cure
Curing effects of viscosity A curing thermoset polymer has a conversion or cure dependent viscosity that increases as the molecular weight of the reacting polymer increases. For vinyl ester, whose curing history is shown in Fig. 5.10 [3], the viscosity behaves as shown in Fig. 5.11 [3]. Hence, a complete model for viscosity of a reacting polymer must contain the effects of strain rate, γ, ˙ temperature, T , and degree of cure, c, such as η = η(γ, ˙ T, c).
(5.14)
There are no generalized models that include all these variables for thermosetting polymers. However, extensive work has been done on the viscosity of polyurethanes [4, 5] used in the reaction injection molding process. An empirical relation that models the viscosity of these mixing-activated polymers, given as a function of temperature and degree of cure, is written as η = η0 eE/RT
cg cg − c
c1 +c2 c (5.15)
where E is the activation energy of the polymer, R is the ideal gas constant, T is the temperature, c g is the gel point1 , c the degree of cure, and c 1 and c2 are constants that fit the experimental data. Figure 5.12 shows the viscosity as a function of time and temperature for a 47% MDI-BDO P(PO-EO) polyurethane. 1 At the gel point, the cross-linking creates a closed network, at which point it is said that the molecular weight goes to infinity.
5.1 Basic Concepts and Terminology
135
100 90 °C 50 °C 30 °C
Viscosity (Pa-s)
10
1
0.1
0.01 0.1
1.0
10
100
Time (min)
Figure 5.12: Viscosity as a function of time for a 47% MDI-BDO P(PO-EO) polyurethane at various isothermal cure temperatures
Suspension rheology: Particles suspended in a material, such as in filled or reinforced polymers, have a direct effect on the properties of the final article and on the viscosity during processing. Numerous models have been proposed to estimate the viscosity of filled liquids [6, 7, 8, 9, 10]. Most models proposed are a power series of the form ηf = 1+a1 φ+a2 φ2 +a3 φ3 +..... η
(5.16)
The linear term in Eq. 5.16 represents the narrowing of the flow passage caused by the filler that is passively entrained by the fluid and sustains no deformation as shown in Fig. 5.13. For Particles V
V v(z)
z . γ
Figure 5.13:
. κγ
vf (z)
Schematic diagram of strain rate increase in a filled system
instance, Einstein’s model, which only includes the linear term with a 1 = 2.5, was derived based on a viscous dissipation balance. The quadratic term in the equation represents the first-order effects of interaction between the filler particles. Geisb¨usch suggested a model with a yield stress and where the strain rate of the melt increases by a factor κ as ηf =
τ0 + κη0 (κγ). ˙ γ˙
(5.17)
136
5 Melt Rheology
7 Experimental data a1 = 2.5, a = 14.1 (Guth, 1938)
6
2
f
η /η
5 4 3 2 1 0
10 30 40 20 Volume fraction of filler (%)
50
Figure 5.14: Viscosity increase as a function of volume fraction of filler for polystyrene and lowdensity polyethylene containing spherical glass particles with diameters ranging between 36 μm and 99.8 μm
For high deformation stresses, which are typical in polymer processing, the yield stress in the filled polymer melt can be neglected. Figure 5.14 compares Geisb¨usch’s experimental data to Eq. 5.16 using the coefficients derived by Guth [9]. The data and Guth’s model seem to agree well. A comprehensive survey on particulate suspensions was recently given by Gupta [11] and on short-fiber suspensions by Milliken and Powell [12]. Complex modulus: The complex modulus G ∗ is the overall resistance to deformation of a material, regardless of whether that deformation is recoverable (elastic) or non-recoverable (viscous). Complex viscosity: The complex viscosity is a ratio of complex modulus to angular frequency, usually denoted by the symbol η ∗ . Dynamic storage modulus: The dynamic storage modulus is the contribution of elastic (solid-like) behavior to the complex modulus, usually denoted by the symbol G . Dynamic loss modulus: The dynamic loss modulus is the component of the complex modulus that is out of face, usually denoted by the symbol G . Compliance: The compliance, J, is the ratio of strain to stress. Viscoelastic memory effects: When a polymer melt is deformed, either by stretching, shearing, or often by a combination of the above, the polymer molecules are stretched and untangled. In time, the molecules try to recover their initial shape, in essence getting used to their new state of deformation. If the deformation is maintained for a short period of time, the molecules may go back to their initial position, and the shape of the melt is fully restored to its initial shape. Here, it is said that the molecules remembered their initial position.
5.1 Basic Concepts and Terminology
137
However, if the shearing or stretching goes on for an extended period of time, the polymer cannot recover its starting shape, in essence forgetting the initial positions of the molecules. The time it takes for a molecule to fully relax and get used to its new state of deformation is referred to as the relaxation time, λ. A useful parameter often used to estimate the elastic effects during flow is the Deborah number 2, De. The Deborah number is defined by De =
λ , tprocess
(5.18)
where tprocess is a characteristic process time. For example, in an extrusion die, a characteristic process time can be defined by the ratio of characteristic die dimension in the flow direction and average speed through the die. A Deborah number of zero represents a viscous fluid 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 extrudate dimension deviations or irregularities such as extrudate swell, shark skin, or even melt fracture. Melt fracture: This phenomenon appears in the form of waves in the extrudate when the polymer is extruded at high speeds and is not allowed to relax. a)
b)
c)
d)
Figure 5.15:
Various shapes of extrudates under melt fracture [14]
This phenomenon is sometimes referred to as shark skin and is shown for a high-density polyethylene in Fig. 5.15a [13]. It is possible to extrude at such high speeds that an intermittent separation of melt and inner die walls occurs, as shown in Fig. 5.15b. This phenomenon is often referred to as the stick-slip effect or spurt flow 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. If the speed is further increased, a helical geometry is extruded as 2 From the Song of Deborah, Judges 5:5 – "The mountains flowed before the Lord." M. Rainer is credited for naming the Deborah number; Physics Today, 1, (1964).
138
5 Melt Rheology
shown for a polypropylene extrudate in Fig. 5.15c. Eventually, the speeds are so high that a chaotic pattern develops, such as the one shown in Fig. 5.15d. This well-known phenomenon is called melt fracture. The shark skin effect is frequently absent, and spurt flow seems to occur only with linear polymers. Table 5.1 presents estimated critical melt fracture stresses for various polymers. Table 5.1:
Critical shear stress for flow instabilities prediction for some polymers Material
Critical shear stress, (MPa)
HDPE
0.145
LLDPE
0.145
LDPE
0.130
PP
0.130
PS
0.130
Normal stresses: 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 τxy = η(γ) ˙ γ˙ xy . (5.19) Measurable normal stress differences, N 1 = τxx -τyy and N2 = τyy -τzz are referred to as the first and second normal stress differences.
Reduced first normal stress difference Log(ψ /a T2 ) (Pa-s 2 )
7 Experiments done between o115 and 210oC 115 C 130 o C 150 o C 170 o C 190 o C 210 o C
6
5
4
3
2
1 -4
-3
-2
-1
0
1
2
3
4
. Reduced shear rate, log aT γo (s-1)
Figure 5.16: Reduced first normal stress difference coefficient for a low-density polyethylene melt at a reference temperature of 150 ◦ C
The first and second normal stress differences are material dependent and are defined by 2 N1 = τxx -τyy = -Ψ1 γ˙ xy
N2 = τyy -τzz =
2 -Ψ2 γ˙ xy .
(5.20) (5.21)
5.1 Basic Concepts and Terminology
139
Reduced viscosity, Log(η/aT) (Pa-s)
7 115 o C 130 o C 150 o C 170 o C 190 o C 210 o C
6
5
4
3
2 1 -4
Figure 5.17: 150 ◦ C
-3
-2
-1 0 1 2 . Reduced shear rate, log aT γo (s-1)
3
4
Reduced viscosity for a low-density polyethylene melt at a reference temperature of
The material functions, Ψ 1 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. 5.20 and 5.21. Figure 5.16 presents the first normal stress difference coefficient for the low density polyethylene melt of Fig. 5.17 at a reference temperature of 150 ◦ C. The second normal stress difference is much smaller than the first normal stress difference and is therefore difficult to measure. Material data banks: The rheological behavior of molten polymers are found in material data banks, such as CAMPUS , and are presented in graphs such as shown in Figs. 5.16 and 5.17. Surface tension: Surface tension plays a significant role in the deformation of polymers during flow, especially in dispersive mixing of polymer blends. Surface tension, σ S , between two materials appears as a result of different intermolecular interactions. In a liquid-liquid system, surface tension manifests itself as a force that tends to maintain the surface between the two materials to a minimum. Thus, the equilibrium shape of a droplet inside a matrix, which is at rest, is a sphere. When three phases touch, such as liquid, gas, and solid, we get different contact angles depending on the surface tension between the three phases. Figure 5.18 schematically depicts three different cases. In case 1, the liquid perfectly wets the surface with a continuous spread, leading to a wetting angle of zero. Case 2, with moderate surface tension effects, shows a liquid that has a tendency to flow over the surface with a contact angle between zero and π/2. Case 3, with a high surface tension effect, is where the liquid does not wet the surface, which results in a contact angle greater than π/2. In Fig. 5.18, σ S denotes the surface tension between the gas and the solid, σ l the surface tension between the liquid and the gas, and σ sl the surface tension between the solid and liquid. The wetting angle can be measured using simple techniques such as a projector, as shown schematically in Fig. 5.19. This technique, originally developed by Zisman, can be used in
140
5 Melt Rheology
Case 1
σl
Case 2
σs
φ
σs,l
Case 3
Figure 5.18: effects
Schematic diagram of contact between liquids and solids with various surface tension
φ Micrometer Syringe Drop Magnifying apparatus xy-translator Optical bench
Figure 5.19:
Schematic diagram of apparatus to measure contact angle between liquids and solids
cosφ
1.0
0.5
0
Figure 5.20:
σc
σl
Contact angle as a function of surface tension
the ASTM D2578 standard test. Here, surface tension, σ l is applied to a film. The measured values of cos φ are plotted as a function of surface tension σ l , as shown in Fig. 5.20, and extrapolated to find the critical surface tension, σ c , required for wetting. For liquids of low-viscosity, a useful measurement technique is the tensiometer, schematically represented in Fig. 5.21. Here, the surface tension is related to the force it takes to pull a platinum ring from a solution. Surface tension for selected polymers are listed in Table 5.2, for some solvents in Table 5.3, and between polymer-polymer systems in Table 5.4. There are many areas in polymer processing and in engineering design with polymers where surface tension plays a significant role. These areas are the mixing of polymer blends, adhesion, treatment of surfaces to make them non-adhesive, and sintering. During manufacturing, it is often necessary to coat and crosslink a surface with a liquid adhesive or bonding
5.1 Basic Concepts and Terminology
141
material. To enhance adhesion it is often necessary to raise surface tension by oxidizing the surface, by creating COOH-groups, using flames, etching, or releasing electrical discharges. Table 5.2:
Typical surface tension values of selected polymers at 180 ◦ C Polymer
σs (N/m)
∂σs /∂T (N/m/K)
0.0290
-
0.035
-5.7 × 10-5
Polyethylene teraphthalate (290 C)
0.045
-6.5 × 10-5
Polyisobutylene
0.0234
-6 × 10-5
Polymethyl methacrylate
0.0289
-7.6 × 10-5
Polypropylene
0.030
-5.8 × 10-5
Polystyrene
0.0292
-7.2 × 10-5
Polytetrafluoroethylene
0.0094
-6.2 × 10-5
Polyamide resins (290 ◦ C) Polyethylene (linear) ◦
Level Force measuring device
Ring
Fluid
Figure 5.21:
Schematic diagram of a tensiometer used to measure surface tension of liquids
Table 5.3:
Surface tension for several solvents Solvent n-Hexane
Surface tension - σs (N/m) 0.0184
Formamide
0.0582
Glycerin
0.0634
Water
0.0728
142
5 Melt Rheology
Table 5.4:
Surface tension between polymers Polymers PE-PP
σs (N/m)
∂σs /∂T (N/m/K)
T (◦ C)
0.0011
-
140 -5
PE-PS
0.0051
2.0×10
180
PE-PMMA
0.0090
1.8 × 10-5
180
PP-PS
0.0051
-
140
PS-PMMA
0.0016
1.3 × 10-5
140
This is also the case when enhancing the adhesion properties of a surface before painting. On the other hand, it is often necessary to reduce adhesiveness of a surface, such as required when releasing a product from the mold cavity or when coating a pan to give it nonstick properties. A material often used for this purpose is polytetrafluoroethylene (PTFE), mostly known by its tradename Teflon. 5.2 CONSTITUTIVE MODELS A constitutive model is a mathematical expression that relates stress, strain, stress rate, strain rate, pressure, and temperature, just to name a few. The most common constitutive model relates stress to strain rate and temperature. The relationship between stress and strain rate is a viscosity function that typically incorporates the dependence of other factors, such as temperature, pressure, and time. It can be stated that the viscosity is a complex function that depends on factors including: • Macromolecular characteristic of the polymer, such as chain branches and rigidity • Molecular weight distribution • Shear rate • Temperature • Pressure • Time • Voltage in the case of electro-rheological fluids • Magnetic field in case of magneto-rheological fluids Although several viscosity functions are presented in the literature, in this book the Newtonian model, the Power law and Bird-Carreau-Yasuda model were selected because they can be easily related to most polymer flows that involve shear. For non-isothermal and nonisobaric problems, models that relate the viscosity to temperature and pressure were also included. Finally, for complex flows, such as those that involve viscoelastic and elongational effects, the Phan-Thien and Tanner multimode model is presented.
5.2 Constitutive Models
143
5.2.1 Newtonian Model The Newtonian model was developed by Newton in 1687 and predicts a viscosity independent of the shear rate (i.e., the shear stress is a linear relationship of shear rate). Although this model is easy to apply, it can be used only for the beginning of the viscosity curve at low shear rates. Mathematically, the Newtonian model can be expressed as τ , γ˙
μ=
(5.22)
where μ is the shear viscosity (Pa· s), γ˙ is the shear rate (s -1 ), and τ is the stress (Pa). Note that for Newtonian fluids the viscosity is represented by μ. 5.2.2 Power Law Model This model, also known as the Ostwald and de Waale model, is easy to use and can represent the pseudoplastic behavior of polymer melts. The term pseudoplastic was introduced by Williamson and Ostwald in 1925 to describe the behavior of the fluids that suffer a reduction of viscosity with an increase of shear rate. It can be applied for the pseudoplastic or shear thinning region because for low shear rates the prediction of viscosity is very high. Mathematically, the Power law model can be expressed as η = m · γ˙ n-1 ,
(5.23)
where η is the shear viscosity (Pa·s), γ˙ is the shear rate (s -1 ), m is the consistency factor (P a · sn ), and n is the Power law index. For polymers the Power law index n is typically between 0.2 and 0.9. The deviation of n from the unity is taken as a measurement of the non-Newtonian behavior. For values of n > 1, the fluid behaves as a dilatant, or shear thickening fluid. Table 5.5 presents a list of typical Power law and consistency indices for common thermoplastics. Table 5.5:
Power law and consistency indices for common thermoplastics
Polymer
m (Pa· sn )
n
T (◦ C)
High-density polyethylene
2.0 × 104
0.41
180
Low-density polyethylene
3
6.0 × 10
0.39
160
Polyamide 66
6.0 × 102
0.66
290
Polycarbonate
2
6.0 × 10
0.98
300
Polypropylene
7.5 × 103
0.38
200
Polystyrene
4
2.8 × 10
0.28
170
Polyvinyl chloride
1.7 × 104
0.26
180
The temperature dependence of the Power-law viscosity can be built into the consistency as m = m0 e-a(T -Tref )
(5.24)
144
5 Melt Rheology
or by introducing a time-temperature superposition shift factor, ˙ n-1 , η = k · aT · (aT · γ)
(5.25)
where for amorphous polymers, the shift factor is best defined using the Williams, Landel, and Ferry model 3 given by aT = 10
C1 (T -Tref ) 2 +(T -Tref )
-C
,
(5.26)
where C1 , C2 are empirical constants obtained by statistical regression of viscosity curve at different temperatures and selecting a particular value of reference temperature (C 1 = 17.44 and C2 = 51.6 when Tref = Tg . If Tref = Tg +45K, C1 = 8.86 and C2 = 101.6), T is the temperature (K), and T ref is the reference temperature (K). A shift commonly used for semicrystalline polymers is the Arrhenius shift, which is written as E0 1 1 , (5.27) Ln(aT ) = R T T0 where E0 is the activation energy (J· mol -1 ) (this constant can be obtained by statistical regression of the viscosity curve at different temperatures and selecting a particular value of reference temperature), T 0 is the reference temperature (K), and R is the universal gas constant (8.314 J· mol -1 · K-1 ).
5.2.3
Bird-Carreau-Yasuda Model
The Bird-Carreau-Yasuda model is more complex and has the advantage of predicting both the Newtonian and the pseudoplastic behavior of polymers, as well as the transition region. Mathematically, the Bird-Carreau-Yasuda model can be expressed as follows η − η0 = [1+|λγ| ˙ a ](n-1)/a , η0 − η∞
(5.28)
where η is the shear viscosity (Pa·s), η 0 is the zero-shear-rate viscosity (Pa·s), η ∞ is the infinite-shear-rate viscosity (Pa·s), λ is the time constant , γ˙ is the shear rate, (s -1 ), and n is the Power law index. In the original Bird-Carreau model, the constant a = 2. In many cases, the infinite-shearrate viscosity is negligible, reducing Eq. 5.28 to a three parameter model. Equation 5.28 was modified by Menges, Wortberg, and Michaeli to include a temperature dependence using a WLF relation. The modified model, which is used in commercial polymer data banks, is written as: A · aT η= , (5.29) [1+B γa ˙ T ]C 3 Often
referred to as WLF equation.
5.2 Constitutive Models
145
where aT is the temperature shift factor (A) is the viscosity constant (Pa·s -C ), B is the time constant (s), and C is the shear thinning index. The shift a T , for the best-fit reference temperature, T ref , is given by4 Log(aT ) =
8.86(Tref -Tg ) 8.86(T -Tg ) − , 101.6+Tref -Tg 101.6+T -Tg
(5.30)
where aT is the temperature shift factor, T g is the glass transition temperature (K), and T ref is the reference temperature (T ref = Tg + 45K). Table 5.6 presents constants for CarreauWLF (amorphous) and Carreau-Arrhenius models (semi-crystalline) for various common thermoplastics. Table 5.6: Constants for Carreau-WLF (amorphous) and Ccarreau-arrhenius (semicrystalline) models for various common thermoplastics Polymer
A (Pa·s)
B (s)
C
Tref (◦ C)
Tg (◦ C)
T0 (◦ C)
E0 (J/mol)
High-density polyethylene
24,198
1.38
0.60
-
-
200
22,272
Low-density polyethylene
317
0.015
0.61
-
-
189
43,694
Polyamide 66
44
0.00059
0.40
-
-
300
123,058
Polycarbonate
305
0.00046
0.48
320
153
-
-
Polypropylene
1,386
0.091
0.68
-
-
220
427,198
Polystyrene
1,777
0.064
0.73
200
123
-
-
Polyvinyl chloride
1,786
0.054
0.73
185
88
-
-
5.2.4 Pressure Dependence of Viscosity The viscosity of a polymer melt is also dependent on the pressure, which is particularly important in injection molding processes of thin parts where the pressure levels are the highest. The pressure dependence of viscosity can be modeled by using: • Pressure Coefficient – The viscosity at a certain pressure can be predicted based on a viscosity data at another pressure if the coefficient of pressure is known, η(P ) = η(Po ) · eφ·P ,
(5.31)
where η(P ) is the viscosity at a pressure p (Pa· s), φ is the pressure coefficient atm-1 , and η(Po ) is the viscosity at a pressure p o (Pa· s). The pressure coeffcient can be obtained by statistical regression of viscosity data at various pressures. Very little information on the coefficient of pressure has been published so far. Table 5.7 presents the reported data for some particular polymers. 4 This
shift should be used for amorphous polymers.
146
5 Melt Rheology
Table 5.7:
Coefficient of pressure for some particular polymers
Polymer
Temperature, ◦ C
φ, 10-3 atm
180 190 200
6.00 4.00 3.00
-
2.31
PVC for injection molding
Linear low-density Polyethylene (LLDPE)
• Modified WLF model – In addition to the temperature shift, Menges, Wortberg, and Michaeli measured a pressure dependence of the viscosity and proposed the following model, which includes both temperature and pressure viscosity shifts: log η(T, p) − log η0 =
8.86(T -T0 ) 8.86(T -T0 +0.02p) , − 101.6+T -T0 101.6+(T -T0 +0.02p)
(5.32)
where η is the pressure dependent viscosity, η 0 is the reference viscosity, and p is the pressure in bar (the constant 0.02 represents a 2 ◦ C shift per bar). 5.2.5 Phan-Thien and Tanner Multimode Model A great variety of viscoelastic models have been used to describe the viscoelastic behavior of polymer melts, including the model independently developed by Phan-Thien and Tanner (PTT) [15] and by Acierno et al. [16]. The multimode Phan-Thien and Tanner model has been successfully used for modeling the shear, elongational, and oscillatory shear behavior of polymers, as well as for modeling the processing of viscoelastic polymers. Mathematically, the multimode Phan-Thien and Tanner (PTT) 2D viscoelastic model in cylindrical coordinates, can be expressed as
exp(
i λi dτzzi dV dV − 2λi (1 − ξi )τzzi = 2ηi (τzzi +2τrri ))τzzi +λi V ηi dz dz dz
(5.33)
i λi dV dV dτrri +λi (1 − ξi )τrri = −ηi (τzzi +2τrri ))τrri +λi V ηi dz dz dz
(5.34)
exp(
τzz =
N i=1
τzzi
τrr =
N
τrri
(5.35)
i=1
where τzz is the z component of the elongational stress, τ rr is the r component of the elongational stress, V is the velocity in z direction, η i is the elongational viscosity parameter for PTT model, λ i is the relaxation time parameter for PTT model, i is the parameter for PTT model, and ξ i is the parameter for PTT model.
5.3 Rheometry
147
5.3 RHEOMETRY There are various ways to qualify and quantify the properties of the polymer melt in industry. The techniques range from simple analyses for checking the consistency of the material at certain conditions to more complex measurements that evaluate viscosity and normal stress differences. This section includes three such techniques to give the reader a general idea of current measuring methods. 5.3.1 The Melt Flow Indexer The melt flow indexer is often used in industry to characterize a polymer melt and as a simple and quick means of quality control. It takes a single point measurement using standard testing conditions specific to each polymer class on a ram-type extruder or extrusion plastometer as shown in Fig. 5.22. Weight
Thermometer
Polymer
Capillary
Figure 5.22:
Schematic diagram of an extrusion plastometer used to measure the melt flow index
The standard procedure for testing the flow rate of thermoplastics using an extrusion plastometer is described in the ASTM D1238 test as presented in Table 5.8. During the test, a sample is heated in the barrel and extruded from a short cylindrical die using a piston actuated by a weight. The weight of the polymer in grams extruded during the 10-minute test is the melt flow index (MFI) of the polymer.
148
5 Melt Rheology
Table 5.8: Standard methods of measuring melt flow index (MFI): melt flow rate (MFR), melt volume rate (MVR), and flow rate ratio (FRR)(after Shastri) Standard
ISO 1133
ASTM D1238 - 98
Specimen
Powder, pellets, granules, or strips of films
Powder, pellets, granules, strips of films, or molded slugs
Conditioning
In accordance with the material standard, if necessary
Check the applicable material specification for any conditioning requirements before using this test. See practice D618 for appropriate conditioning practices.
Apparatus
Extrusion plastometer with a steel cylinder (115 – 180) mm (L) x 9.55 ±0.025 mm (D), and a die with an orifice of 8.000 ±0.025 mm (L) x 2.095 ±0.005 mm (D)
Extrusion plastometer with a steel cylinder 162 mm (L) x 9.55 ±0. 008 mm (D), and a die with an orifice of 8.000 ±0.025 mm (L) x 2.0955 ±0.0051 mm (D)
Test procedures
Test temperature and test load as specified in Part 2 of the material designation standards, or as listed in ISO 1133. Some examples are: PC (300 ◦ C /1.2 kg) ABS (220 ◦ C /10 kg) PS (200 ◦ C /5 kg) PS-HI (200 ◦ C /5 kg) SAN (220 ◦ C /10 kg) PP (230 ◦ C /2.16 kg) PE (190 ◦ C / 2.16 kg) POM (190 ◦ C /2.16 kg) PMMA (230 ◦ C /3.8 kg) Charge ⇒ within 1 min Preheat ⇒ 4 min Test time ⇒ last measurement not to exceed 25 min from charging. Procedure A – manual operation using the mass and cuttime intervals shown in the following:
Test temperature and test load as specified in the applicable material specification, or as listed in D1238. Some examples are: PC (300 ◦ C /1.2 kg) ABS (230 ◦ C /10 kg) PS (200 ◦ C /5 kg) PS-HI (200 ◦ C /5 kg) SAN (220 ◦ C /10 kg) PP (230 ◦ C /2.16 kg) PE (190 ◦ C / 2.16 kg) POM (190 ◦ C /2.16 kg) Acrylics (230 ◦ C /3.8 kg) Charge ⇒ within 1 min Preheat ⇒ 6.5 min Test time ⇒ 7.0 ±0.5 min from initial charging. Procedure A – manual operation using the mass and cuttime intervals shown in the following: Continued on next page
5.3 Rheometry
149
Standard
Values and units
ISO 1133
ASTM D1238 - 98
MFR; Mass; Time 0.1 to 0.5g/10min; 3–5g ; 4min >0.5 to 1g/10min; 4–5g; 2min >1 to 3.5g/10min; 4–5g; 1min >3.5 to 10 g/10 min 6– 8g; 30s > 10g/10min; 6–8g; 5-15s Procedure B – automated time or travel indicator is used to calculate the MFR (MVR) using the mass as specified above in Procedure A for the predicted MFR
MFR; Mass; Time 0.15 to 1g/10min; 2.5–3g; 6 min >1 to 3.5g/10min; 3–5g; 3 min >3.5 to 10g/10min; 4–8g; 1 min >10 to 25g/10min; 4–8g; 30s > 25g/10min; 4–8g; 15s Procedure B – MFR (MVR) is calculated from automated time measurement based on specified travel distances, < 10 MFR ⇒ 6.35 ±0.25 mm. > 10 MFR ⇒ 25.4 ±0.25 mm. and using the mass as specified above for the predicted MFR
MFR ⇒g/10min MVR ⇒cm3 /10min
MFR ⇒ g/10min MVR ⇒ cm3 /10min FRR ⇒ Ratio of the MFR (190/10) by MFR (190/2.16) (used specifically for PE)
INDUSTRIAL APPLICATION 5.1
Industrial Application of Weathering of an Exterior Polyethylene Application A failed polyethylene component exposed to the elements was analyzed to determine the cause of failure (Figure 5.23). Most likely, the cause of failure was degradation due to UV radiation. UV rays can lead to molecular chain sission, often leading to a significant reduction in molecular weight.
Figure 5.23:
Failed polyethylene part
150
5 Melt Rheology
Abs
Part surface
Abs
Part core
Abs
Polyethylene 0.6 0.4 0.2 4000
3000
2000
-1
Wavelength (cm )
1000
500
Figure 5.24: FTIR Spectrums of the surface of the failed part (top), inner core (center) and reference polyethylene (bottom)
A common test performed to detect a loss in molecular weight is measuring changes in melt flow index. For the failed parts the melt flow index was measured to be above 150 g/10 minutes. The material specified for this application was an HDPE with a melt flow index of 15 g/10 minutes. This large difference between specified and tested MFI is due to a significant loss in molecular weight as a result of UV degradation. Another analytical test that can be performed to detect UV degradation is a Fourier Transform Infrared Spectroscopy (FTIR). FTIR spectroscopy was performed at the surface and core of the HDPE sample, and is shown in Fig. 5.24. The FTIR shows the typical spectral results expected for polyethylene. However, the spectrum shows two additional absorption bands (one between 1750 cm -1 and 1700 cm -1 , and another between 1300 cm -1 and 1100 cm -1 ) that indicate the formation of carbonyls and byproducts associated with oxidation, a result of UV degradation. The FTIR performed at the surface of the part shows stronger absorption bands compared to the FTIR at the core. Therefore, as expected, the level of oxidation at the surface is much higher than the oxidation at the core of the part. If the reduction in properties had been caused by oxidation during processing the part would have exhibited uniform degradation throughout the thickness of the part. 5.3.2 Capillary Viscometer The most common and simplest device for measuring viscosity is the capillary viscometer. Its main component is a straight tube or capillary, and it was first used to measure the viscosity of water by Hagen and Poiseuille. A capillary viscometer has a pressure driven flow for which the velocity gradient or strain rate and also the shear rate will be maximum at the wall and zero at the center of the flow, making it a non-homogeneous flow. Since
5.3 Rheometry
151
pressure driven viscometers employ non-homogeneous flows, they can only measure steady shear functions such as viscosity, η(γ). ˙ However, they are widely used because they are relatively inexpensive to build and simple to operate. Despite their simplicity, long capillary viscometers provide the most accurate viscosity data available. Another major advantage is that the capillary viscometer has no free surfaces in the test region, unlike other types of rheometers, such as the cone and plate rheometers, which we will discuss in the next section. When the strain rate dependent viscosity of polymer melts is measured, capillary viscometers may provide the only satisfactory method of obtaining such data at shear rates >10 s-1 . This is important for processes with higher rates of deformation such as mixing, extrusion, and injection molding. Other advantages of the capillary viscometer include. • Capillary flows and geometries are very similar to those encountered in real processing equipment. • A capillary viscometer can be adapted to on-line measurement. • The system allows the study of flow anomalies such as extrudate swell, melt fracture, or stick-slip conditions. • A capillary viscometer can be used to study the pressure dependence of viscosity Heater
Insulation
Pressure transducer
L Polymer sample
Extrudate R
Figure 5.25:
Schematic diagram of a capillary viscometer
As shown in Fig. 5.25, in a capillary viscometer the material is fed into a cylinder where the temperature is maintained within a very narrow range (about T ±0.5 ◦ C). Once the material is molten, the piston traveling at a well controlled speed pushes the material through the capillary. The pressure is measured at the inlet of the circular capillary, and for rectangular capillaries the pressure can be measured inside of the capillary. Using the piston speed, the dimensions of the piston, and the capillary, the apparent shear rate can be calculated; and using the pressure and the dimensions of the capillary, the apparent shear stress can be computed. With the apparent shear rate and shear stress, the apparent viscosity can be computed. Tests at different piston speeds and temperatures can be carried out to obtain an apparent viscosity curve. Subsequent corrections on shear rate and shear stress allow the prediction of the actual viscosity curve. The capillary extrudate can be collected to observe the range of flow anomalies when they occur.
152
5 Melt Rheology
The capillary viscometer includes the following instrumentation: • • • •
A piston and its speed control A cylinder with temperature control An auxiliary extruder and purge valve (optional) A capillary set
There are three types of capillaries in use today: • Circular capillary – This capillary works best for higher shear rates, so it is useful to define the pseudoplastic behavior of polymer melts. The geometrical parameters required to obtain the viscosity data are the inside diameter D (or radius R) and the length L. • Rectangular capillary – This capillary is used for lower shear rates, so it is useful to measure the Newtonian plateau of a polymer melt. This capillary has the advantage that the pressure can be measured directly inside the capillary. Therefore, inlet pressure correction, such as Bagley corrections, are not required. This type of capillary is easier to clean up. The required geometrical parameters to obtain the viscosity data are the internal height h, the width b, and the length L. To minimize the entrance effects, a b/h ratio of 15:1 is normally used. To obtain a complete viscosity curve, measurements with circular and rectangular capillaries are usually done. • Annular capillary – This capillary allows the alteration of the length and the radius by changing only the core and does not have entrance effects. The main disadvantages are that it is difficult to control the temperature of the core and that a considerably higher amount of polymer melt is required. The required geometrical parameters to obtain the viscosity data are the inside diameter D i (or inside radius R i ), the outside diameter Do (or outside radius R o ), and the length L. Annular capillaries are fabricated with a diameter of 20 mm and ratio D i /(Do − Di ) higher than 20. For accurate measurements a piston displacement control is needed. Three possible piston displacement controls exist. These are: • Constant speed control – This control provides a constant piston speed, if the seal between piston and cylinder is adequate. In this alternative the displacement of the piston is controlled while the pressure is registered. Two possible constant speed controllers can be used: – Mechanical – This controller uses a screw that moves axially at a constant linear speed – Hydraulic – This controller uses a very precise hydraulic system to push the piston at a carefully controlled speed • Constant pressure control – Here, pressurized gas pushes the piston. With the help of a pressure sensor located at the inlet (or inside) of the capillary, a constant pressure can be obtained and the volumetric flow can be measured. • Constant volume control – this technology uses a gear pump or an extruder to control the flow of melt while the pressure is measured. Although this alternative nears actual processing conditions, controlling a constant flow is difficult to attain.
5.3 Rheometry
153
The extruder is often a desirable and recommended component of the capillary viscometer to avoid air entrapment and to obtain thermal homogeneity in the melt. A manual load of the cylinder can be done. The extruder delivers the molten polymer to be fed to the cylinder by means of a three-way valve. Corrections: Because of geometrical constraints, the pressure sensor cannot be placed inside the circular capillary. The deformation at the inlet of the capillary leads to pressure losses that require a correction of the shear stress. The correction of the inlet pressure loss is normally called the Ryder-Bagley correction. In the calculation of the shear rate, Newtonian equations are used, so it is necessary to correct for the non-Newtonian behavior of polymer melts. The correction of the non-Newtonian behavior is called the WeissenbergRabinowitsch correction. Ryder-Bagley Correction: For the Ryder-Bagley correction, measurements of pressure for several capillary lengths must be done. At least three capillary lengths are normally used. When the sensor pressures plotted as a function of the capillary lengths at each shear rate, straight lines (or parabolic lines for higher shear rates) are obtained (see Fig. 5.26). The intercept with the y-axis is the Ryder-Bagley correction and has to be deducted from the measured pressure to obtain the real shear stress. The corrected shear stress for a circular capillary can be calculated using τ=
(ΔP − ΔPBagley ) · R , 2·L
(5.36)
where τ is the corrected shear stress (Pa), ΔP is the measured pressure (Pa), ΔP Bagley is the Ryder-Bagley correction (Pa), R is the capillary internal radius (mm), and L is the capillary length (mm).
Figure 5.26:
Ryder-Bagley correction for polypropylene at different shear rates
154
5 Melt Rheology
Table 5.9:
Coefficient of pressure for some particular polymers Circular capillary
Rectangular capillary
Shear rate for a Newtonian fluid 4·V˙ γ˙ ap = π·R 3
Shear rate for a Newtonian fluid 6·V˙ γ˙ ap = b·h 2
Weissenberg-Rabinowitsch correction
Weissenber-Rabinowitsch correction
First point (i = 1) γ˙ wap2 γ˙ w1 = 34 γ˙ wap1 + 14 τwap1 τwap
First point (i = 1) γ˙ 2 γ˙ w1 = 23 γ˙ wap1 + 13 τwap1 wap τw
Intermediate point γ˙ −γ˙ wapi-1 i+1 γ˙ w1 = 34 γ˙ wapi + 14 τw1 wap τw −τw
Intermediate point γ˙ −γ˙ wapi-1 i+1 γ˙ w1 = 23 γ˙ wapi + 13 τw1 wap τw -τw
Last point (i = n) ˙ γ˙ wn = γ˙ wn V˙Vn
Last point (i = n) ˙ γ˙ wn = γ˙ wn V˙Vn
2
i+1
i-1
2
n-1
i+1
i-1
n-1
Weissenberg-Rabinowitsch Correction: As mentioned earlier, Newtonian equations are used in the calculation of the shear rate. Because polymer melts are non-Newtonian, a correction must be done that takes into account the shear thinning behavior. This correction is the socalled Weissenberg-Rabinowitsch correction. Table 5.9 presents the equations to calculate the shear rate for Newtonian fluids and the Weissenberg-Rabinowitsch correction for circular and rectangular capillaries. The standardized techniques used to measure rheological properties of polymeric materials by means of a capillary viscometer are the ISO 11443 and the ASTM D3835 tests. Both tests are presented in Table 5.10. The ASTM D5099 test is used to measure rheological properties of rubber materials using capillary viscometry. The ASTM D5099 test is presented in Table 5.11. Table 5.10: Standard test method for determination of properties of polymeric materials by means of a capillary viscometer Standard
ISO 11443:1995
ASTM D3835-02
Abstract
ISO 11443:2005 specifies methods for determining the fluidity of polymer melts subjected to shear stresses at rates and temperatures approximating to those arising in plastics processing.
This test method covers measurement of the rheological properties of polymeric materials at various temperatures and shear rates common to processing equipment. Continued on next page
5.3 Rheometry
155
Standard
ISO 11443:1995
ASTM D3835-02
Specimen
Plastic melt forced through a capillary or slit die of known dimensions. A small representative sample is taken from the product to be tested.
The test specimen may be in any form that can be introduced into the bore of the cylinder such as powder, beads, pellets, strips of film, or molded slugs. In some cases it may be desirable to preform or pelletize a powder.
Apparatus
A heatable barrel, the bore of which is closed at the bottom end by an exchangeable capillary or slit die. The test pressure shall be exerted on the melt contained in this barrel by a piston, a screw or gas pressure
A capillary viscometer, the barrel, the capillary with a smooth straight bore, and the piston.
Test procedures
The polymer is introduced in the barrel preheated and forced through the capillary at a predetermined piston velocity. The pressure observed is registered with a pressure transducer. The pressure registered and the geometry of the capillary is used to calculate the shear stress (see equations in Table 5.9). The piston velocity and geometry of the barrel and capillary are used to calculate shear rate. For circular capillaries two corrections are performed, the RyderBagley and the WeissenbergRabinowitsch. For rectangular capillaries only the Weissenberg-Rabinowitsch correction is performed.
The polymer is introduced in the barrel preheated and forced to the capillary at a predetermined piston velocity. The pressure observed is registered with a pressure transducer. The pressure registered and the geometry of the capillary is used to calculate the shear stress.
Shear stress (Pa), Shear rate (s-1 ),Viscosity (Pa·s)
Shear stress, (Pa),Shear rate (s-1 ), Viscosity (Pa·s)
Values and Units
The piston velocity and geometry of the barrel and capillary are used to calculate the shear rate. For circular capillaries two correction are performed, the Ryder-Bagley and the Weissenberg-Rabinowitsch. For rectangular capillaries only the Weissenberg-Rabinowitsch correction is performed.
156
5 Melt Rheology
Table 5.11: Standard test methods for rubber-measurement of processing properties using capillary rheometry Standard
ASTM D5099-93(2003)
Scope
This test methods describes how capillary rheometry may be used to measure the rheological characteristics of rubber. Two methods are covered, Method A, which uses a piston type capillary viscometer, and Method B, which uses a screw extrusion type capillary viscometer. The two methods have important differences, as outlined by the test. The test methods cover the use of a capillary viscometer for the measurement of the flow properties of thermoplastic elastomers, unvulcanized rubber, and rubber compounds. These material properties are related to factory processing. Since the piston type capillary viscometers impart only a small amount of shearing energy to the sample, these measurements directly relate to the state of the compound at the time of sampling. Piston capillary viscometer measurements will usually differ from measurements with a screw extrusion type rheometer, which imparts shearing energy just before the rheological measurement. The capillary viscometer measurements for plastics are described in test method ASTM D3835.
Specimen
Massed specimen of raw or compounded unvulcsanized rubber for test method A -Piston extrusion capillary viscometer. Raw rubber or unvulcanized elastomeric compound formed into sheets on a two-roll mill for test method B - Screw extrusion capillary viscometer
Apparatus
A piston type capillary viscometer for test method A A screw extrusion capillary viscometer for test method B
Test procedures
Test method A - Piston extrusion capillary viscometer Unvulcanized rubber compound is placed in a temperature controlled cylinder fitted at one end with a conical transition section and a standard capillary die. The sample is driven through the die with the help of the piston while measuring or controlling the rate of extrusion and the pressure on the sample at the entrance of the die. Test method B - Screw extrusion capillary viscometer Unvulcanized rubber compound compound is formed into sheets on a two-roll mill. Strips cut from these sheets are fed to the extruder whose barrel is equipped with a temperature control. The end of the extruder is equipped with a transition conical section and a capillary die. A pressure transducer and temperature measuring device are placed in the chamber before the die. The rate of extrusion is calculated from the amount of extrudate collected over a timed interval. The rate of extrusion is controlled by adjusting the drive speed. Continued on next page
5.3 Rheometry
157
Standard
ASTM D5099-93(2003)
Values and Units
Viscosity curve (apparent and corrected) in log-log graph Corrected shear stress at 500 s-1 Corrected shear stress at 1000 s-1 Corrected viscosity at 500 s-1 , Corrected viscosity at 1000 s-1 Shear sensitivity, N Entrance effect, E
5.3.3 Rotational Rheometry A rotational rheometer is a particular type of rheometer in which the shear is produced by a drag flow between a moving part and a fixed one, including the following geometries: plateplate, cone-plate, and concentric cylinders. The main features of a rotational rheometer are the following: • The rotational rheometer can measure rheological properties under transient and steady state conditions. • It can reach the lowest shear rates of all rheometers. With this equipment it is possible to obtain shear rates typically from 10 -6 to 2.5 · 102 s-1 , so it is useful for macromolecular characterization (molecular weight distribution, relaxation spectra, and chain branching). • It can be used to measure the normal force in a polymer melt. • It offers a high degree of versatility because the following type of tests can be carried out: dynamic test (oscillation), flow test (rotation), static test, temperature sweep, torque sweep, frequency sweep, and time sweep. • It is used in conjunction with the capillary viscometer to obtain the complete viscosity curve. Rotational rheometers are used when more complex properties, such as normal stresses, are sought. There are two main types of rotational rheometers: the controlled rate rheometer (CRR), in which the strain or the shear strain is imposed and the stress is measured and the controlled stress rheometer (CSR), in which the stress is imposed and the strain or the shear rate is measured. Advances in rotational rheometer instrumentation have made it possible to have systems where both controlled rate and controlled stress can be programmed. Rotational rheometers include a temperature controller (which can be electrical heated plates, a Peltier system or an environmental test chamber), a test geometry (being the most common – plate-plate, cone-plate and concentric cylinders), a magnetic induction motor (being the most common a drag cup motor), an angular displacement measurement device (being the most typical an optical encoder), an electronic system to measure or control the torque, a mechanical frame, and a computer-based data acquisition and processing unit [17]. Parallel-plate rheometer: A parallel plate rheometer, schematically depicted in Fig. 5.27, is the geometrically simplest rotational rheometer, but mathematically it is more complex to analyze than its counterpart, the cone-and-plate rheometer.
158
5 Melt Rheology
Force
Torque
Ω θ
h R
Figure 5.27:
Schematic diagram of a parallel-plate rheometer
The plate-plate rheometer is sometimes the preferred system because of the following advantages: • Easy sample preparation of viscous materials and soft solids. • The shear rate can be easily changed by programming different rotational speeds or by adjusting the gap between plates or by changing the frequency. • Higher shear rates can be obtained before edge effects appear. • In conjunction with cone-plate geometry, the normal stress can be measured. • This geometry is preferred for viscous melts when small shear rates are required. The following equations are normally used in the flow mode: γ˙ R = M Ω
(5.37)
γR = M ϕ
(5.38)
where M is the geometric factor (R/h), h is the gap (mm), R is the external radius of the plate (mm), and ϕ is the deflection angle. Ω=
2π · N 60
(5.39)
where Ω is the angular speed (s -1 ) and N is the rotor speed (rpm). τ = Md · A · (
3+n ) 4
(5.40)
where Md is the torque (N· m), A is the geometric factor 2/R 3 (m-3 ), and n is the Power law exponent (Weissenberg correction). N1 -N2 = (
2Fn 1 dlnFn )(1+ ) πR2 2 dlnγ˙ R
(5.41)
where N1 -N2 is the second normal stress difference (Pa), and F n is the normal force (N).
5.3 Rheometry
159
Cone-plate geometry: The cone-plate rheometer is sometimes the preferred system because of the following advantages: • • • •
This rheometer allows the measurement of normal stresses. Homogeneous strain and simple equations. Useful for measurement of linear viscoelasticity, G(t, γ). Normally used for intermediate viscosity ranges. High-viscosity measurements are limited by the elastic problems in the borders and low-viscosity measurements are limited by inertial effects and sample loss in the borders. Torque
Force Ω
φ
θ
θo
Fixed plate Pressure transducers
R
Figure 5.28:
Schematic diagram of a cone-plate rheometer
The following equations are used in the flow mode: Ω β ϕ γR = β where Ω is the angular speed (s -1 ), β is the cone angle, and ϕ the deflection angle. γ˙ R =
3Md 2πR3 component) (Pa), and M d is the torque (N· m). τ=
where τ is the shear stress (τΦΘ
(5.42) (5.43)
(5.44)
3+n ) (5.45) 4 where Md is the torque (N· m), A is the geometric factor 2/R 3 , (m-3 ), and n is the Power law exponent (Weissenberg correction). τ = Md · A · (
2Fn ) (5.46) πR2 where N1 -N2 is the second normal stress difference (τ ΦΦ -τΘΘ component), (Pa), and F n is the normal force (N). N1 = (
160
5 Melt Rheology
Concentric cylinders geometry The concentric cylinders rheometer (also called the Couette rheometer) has the following characteristics: • • • •
Better for low-viscosity samples (under 100 (Pa·s)) Useful in high shear rates Gravity settling of a suspension has less effect than in a cone and plate rheometer Normal stress is difficult to measure Ω, T
Ri
L
Ro
Polymer
Figure 5.29:
Schematic diagram of a Couette rheometer.
The following equations are used in the flow mode: γ˙ = ϕ γ(R ˙ i) = γ(R ˙ i) =
Θ(Ro +Ri) 2(Ro − Ri)
2Ω Ri 2 (1 − ( R ) ) o 2Ω
n(1 −
2 Ri n (R ) ) o
n=
f or f or
dlnMd d(lnΩ)
(5.47) Ri > 0.99 Ro
0.50 <
Ri < 0.99 Ro
(5.48) (5.49) (5.50)
where Ω is the angular speed (s -1 ), Ro is the outside radius (mm), R i is the internal radius (mm), ϕ is the deflection angle, n is the Power law index (Weissenberg correction), and M d is the torque (N·m). τ=
Md 2πRi2 L
(5.51)
5.3 Rheometry
161
where τ is the shear stress (τΦΘ component) (Pa), M d is the torque (N·m), R i is the internal radius (mm), and L is the length of inside cylinder (mm). Temperature controllers: Because all rheological properties depend on temperature, the success of a rheological measurement starts with a very precise control of the temperature. Modern rheometers include the following types of temperature controls: • Fluid circulators – Fluid circulators use a thermostatic bath that precisely controls the temperature and circulates the fluid in the plates of the rheometer. Typically this control has a temperature range between -40 ◦ C and 250 ◦ C. • Peltier plate – This is the most common temperature control in rheometers and it uses the thermo-electric phenomenon called Peltier effect. The main limitation for a polymer analysis is the temperature range of -20 ◦ C to 200 ◦ C. The Peltier plate is a very accurate control, with a tolerance of about ±0.1 ◦ C, and has typical heating rates up to 20 ◦ C/min. • Electrical heated plates – In this temperature control, one plate is electrically heated. Typically, this control has a temperature range of -130 ◦ C up to 400 ◦ C. The low temperatures could be obtained with a special cooling device. • Thermal chamber – This is an oven that operates by convection and radiation. The main advantage of this system is the possibility of higher temperatures (typically from room temperature up to 1000 ◦ C, with the possibility of starting from -160 ◦ C using a special liquid (nitrogen) cooling device). Typical heating rates are up to 60 ◦ C/min.
0.5
-0.5
Figure 5.30:
Description of the dynamic test in a rotational rheometer
Operation modes: One of the advantages of the rotational rheometer is the versatility that allows different operation modes and tests. Rotational rheometry operation modes are presented in the following. Dynamic Test: These tests are done while subjecting one of the plates to oscillatory motion and by varying the frequency and the amplitude of the oscillation at isothermal conditions (see Fig. 5.30). This operation test can be used to obtain the complex viscosity curve, and its components, as a function of frequency. According to the Cox-Merz rule, in order to have a
162
5 Melt Rheology
complete viscosity curve, the complex viscosity curve can be superimposed with the viscosity curve obtained from a flow test and a capillary viscometer. This operation test obtains important viscoelastic functions, such as dynamic storage modulus G (ω) and dynamic loss modulus G (ω). Flow Test: In this test, one of the plates is subjected to a continuous rotational motion, enabling viscosity measurements as a function of the shear rate under realistic continuous flow. The main limitations are the centrifugal forces and border effects. Creep Test: The creep test, also called retardation, is a special test where the fluid is subjected to a stress step (stress controlled mode), and the strain variation during a period of time is registered. Eventually, the stress is released again and the strain recovery is registered. This experiment reflects viscoelastic behavior and the macromolecular characteristics of polymer melt, such as relaxation times. Relaxation Test: In the relaxation test, the fluid is subjected to a strain step (strain-controlled mode), and after some period of time the stress variation as a function of time, is registered. The relaxation test provides important information about the viscoelastic behavior and the macromolecular characteristics of polymer melts, such as stress overshoot. Torque Sweep: This test is useful in identifying the linear viscoelasticity region, which is the region where the compliance curves at different stresses can be super-imposed in a master curve independent of the applied stress. As a rule, the majority of rheological characterizations of a polymeric fluid are done in the linear viscoelastic region, so it is first recommended to determine the stress that limits this region. Frequency Sweep: This is a special dynamic test used to obtain the viscosity curve and important information about the viscoelastic behavior of a polymer melt. A sweep of frequency is normally done from very low to higher possible values without any border effect and under isothermal conditions. Temperature Sweep: In this test, the polymer sample is evaluated at a certain fixed frequency under a temperature program. The temperature sweep is particularly useful to determine some transition temperatures, such as glass transition and curing rate of thermosets and rubbers. The temperature sweep is also used to study curing of thermosets and degradation of polymers. Time Sweep: In this test, the polymer sample is evaluated at a given frequency during a period of time, and the change in the viscosity as a function of time is recorded. This particular test is useful in characterizing the thixotropic and rheopexic fluids. The thixotropic materials are fluids where the viscosity decreases with the time at a fixed shear rate, while rheopexic materials are fluids where the viscosity increases with time at a fixed shear rate. The standardized test to measure complex properties using parallel plate as well as coneand-plate rheometers are the ISO 6721 and ASTM D4440 tests presented in Table 5.12.
5.3 Rheometry
163
Table 5.12: Dynamic mechanical properties - complex shear viscosity using a parallelplate oscillatory rheometer Standard
ISO 6721-10:1999
ASTM D4440-01
Abstract/Scope
Determination of dynamic mechanical properties, such as complex shear viscosity as a function of frequency, strain amplitude, temperature, and time, using a parallel-plate oscillatory rheometer. This part specifies the general principles of a method for determining the dynamic rheological properties of polymer melts at angular frequencies typically in the range 0.01 10 Hz by means of an oscillatory rheometer with a parallel plate test geometry. Frequencies outside this range can be used if edge distortions and anomalies are not observed. The method is used to determine values of the dynamic rheological properties: complex shear viscosity h∗ , dynamic shear viscosity h’, the out-of-phase component of the complex shear viscosity h", complex shear modulus G∗ , shear loss modulus G" and shear storage modulus G".
Dynamic mechanical test to measure rheological properties of thermoplastic resins, such as complex viscosity, tan δ and significant viscoelastic characteristics as a function of frequency, strain amplitude, temperature, and time. This test method is valid for a wide range of frequencies, typically from 0.01 to 100 Hz.
This test method is intended for homogenous and heterogeneous molten polymeric systems and composite formulations containing chemical additives, including fillers, reinforcements, stabilizers, plasticizers, flame retardants, impact modifiers, processing aids, and other important chemical additives often incorporated into a polymeric system for specific functional properties, and which could affect the processability and functional performance. Apparent discrepancies may arise in results obtained under differing experimental conditions. Continued on next page
164
5 Melt Rheology
Standard
ISO 6721-10:1999
ASTM D4440-01
It is suitable for measuring complex shear viscosity values typically up to approximately 10 MPa.s.Test data obtained by this test method are relevant and appropriate for use in engineering design.
Without changing the observed data, reporting in full (as described in this test method) the conditions under which the data were obtained will enable apparent differences observed in another study to be reconciled. Test data obtained by this test method are relevant and appropriate for use in engineering design.
Specimen
In the form of a disc when produced by injection or compression molding or by cutting from sheet. Also, pellets or molten polymer.
A known amount of thermoplastic resin (molten powder or pellet, or solid preform disk) Molten polymer should be both homogeneous and representative.
Apparatus
Two concentric, rigid, circular parallel plates between which the specimen is placed. One of these plates oscillates at a constant angular frequency while the other remains at rest. An angular displacement and a torque measuring device record the strain and the stress during the test.
An apparatus to hold a molten polymer of known volume and dimensions so that the material acts as the elastic and dissipative element in a mechanically driven oscillatory system. The apparatus consists of the test fixtures (polished cone and plate, or parallel plates having either smooth, polished, or serrated surface), oscillatory deformation device, detectors (to determine stress, strain, frequency, and temperature), temperature controller and oven, Nitrogen, or other gas supply for purging purposes.
Test procedures
The specimen is held between the parallel plates and subjected to either a sinusoidal torque (controlled-stress mode) or sinusoidal angular displacement (controlledstrain mode).
Specimen is held between parallel plates or cone and plate and subjected to either a sinusoidal torque (controlled-stress mode) or sinusoidal angular displacement (controlledstrain mode). Continued on next page
5.3 Rheometry
Standard
Values and Units
5.3.4
165
ISO 6721-10:1999
ASTM D4440-01
In the controlled-stress mode the resultant displacement and the phase shift between torque and displacement are registered. In the controlledstrain mode the resultant torque and the phase shift between torque and displacement are registered. The equipment is able to measure important viscoelastic functions of the polymer melts.
In the controlled-stress mode the resultant displacement and the phase shift between torque and displacement are registered. In the controlled-strain mode the resultant torque and the phase shift between torque and displacement are registered. The equipment is able to measure important viscoelastic functions for the polymer melt under consideration.
Torque, angular displacement, angular frequency, shear stress, shear strain, shear storage modulus, shear loss modulus, complex shear modulus, dynamic shear viscosity, out-of-phase component of the complex shear viscosity, complex shear viscosity, and phase shift or loss angle all in SI units.
Dynamic moduli, complex viscosity, and tan δ as a function of the dynamic oscillation (frequency), percent strain, temperature, or time, all given in the standard SI units.
Extensional or Elongational Rheometry
It should be emphasized that the shear behavior of polymers measured with the equipment described in the previous sections cannot be used to deduce the extensional behavior of polymer melts. Extensional rheometry is the least understood field of rheology. Elongational or extensional properties are important when analyzing and understanding fiber spinning, thermoforming, film blowing, film casting, blow molding, and foaming. Several elongational rheometers have been designed to measure the elongational viscosity (or elongational stress) as a function of the elongational rate of deformation at different temperatures. However, the main challenges are how to obtain higher elongational rates, as well as simplifying the measurements, and making them reproducible. When measuring extensional viscosities, we can divide the techniques into direct and indirect methods. Direct measurement of elongational viscosity The following rheometers and elongational techniques, which fall under direct measuring techniques, have been proposed and used in the past [18]. • Extensional methods – These methods are more direct ways to measure the elongational behavior of polymers. Several types of extensional rheometers have been used, the following being the most common:
166
5 Melt Rheology
– Uniaxial extension – In this particular technique, the polymeric sample is stretched in one direction, and the elongational stress is measured under a defined elongation rate. Some examples of this type of rheometer are the Meissner rheometer [18], the vertical buoyancy bath [18], and the Sentmanat extensional rheometer (SER) [19, 20]. A schematic of Meissner’s extensional rheometer incorporating rotary clamps is shown in Fig. 5.31.
Spring εr = ln LA/LR Displacement sensor
Drive motor
LR LA
Sample Lo
Figure 5.31:
Schematic diagram of an extensional rheometer
– Lubricated compression – Another setup that can be used to measure extensional properties without clamping problems and without generating orientation during the measurement is the lubricating squeezing flow, which generates an equibiaxial deformation. To lubricate the material, the plates are usually coated with polydimethylsiloxane (silicone oil). A schematic of this apparatus is shown in Fig. 5.32.
Figure 5.32:
Schematic diagram of squeezing flow
– Biaxial and multiaxial extension – For high-viscosity polymers and rubber, it is possible to use a special rheometer with translating clamps that move in the orthogonal axis [18]. Rheometers have been built in the past that are able to move the sample in all directions, generating an equibiaxial extension of the sample.
5.3 Rheometry
167
– Bubble blowing – With this system, a sheet is clamped between two plates with circular holes and a pressure differential is introduced to deform it and blow the bubble into a test fluid. The pressure applied and deformation of the sheet are monitored over time and related to extensional properties of the material. The radius of the bubble allows the measurement of the extensional rate and the extensional strain, and the pressure difference and the interfacial tension allow one to determine the elongational stress [18]. The bubble blowing system is schematically depicted in Fig. 5.33. This test has been successfully used to measure extensional properties of polymer membranes for blow molding and thermoforming applications.
h
α
Figure 5.33:
R
Schematic diagram of sheet inflation
– Fiber spinning – This method is particularly useful for low-viscosity samples, where the polymer is continuously extruded and stretched by a rotating wheel. The diameter of the fiber as a function of the axial distance can be measured photographically and the force measured by the wheel with a load cell [18, 23, 24].
Figure 5.34:
Schematic diagram of a Rheotens extensional rheometer
168
5 Melt Rheology
– Rheotens – This technique was developed by the company G¨ottfert and consists of a tandem pulley system in which the melt that comes out from the circular capillary is pulled off between two sets of counter-rotating pulleys. A Rheotens rheometer is schematically depicted in Fig. 5.34. One of the pulleys is used to measure the torque [21]. The elongation viscosity as a function of the elongational rate can be determined with the help of software developed by Wagner and coworkers at the IKT, University of Stuttgart, Germany [22]. Indirect measurement of elongational viscosity Indirect methods for measuring elongational viscosity have been reported in the literature. The most popular method uses the pressure drop in sudden flow contraction and the stagnation flow. • Flow stagnation – This technique uses the principle that steady extensional deformations can be created by impinging two liquid streams, such as depicted in Fig. 5.35 [18]. Although with stagnation flows it is only possible to measure steady extensional viscosity, there is great interest in this technique because high elongational rates and low-viscosity samples can be studied.
Stagnant region
Figure 5.35:
Schematic diagram of an impinging flow
• Entrance flows – This method is based on the pressure losses in sudden flow contractions and can be considered as a special case of a flow stagnation technique. The most common methods are the following: – Cogswell’s method – According to the theory developed by Cogswell, the elongational viscosity is obtained from the pressure drop in a sudden flow contraction, such as a capillary with an inlet angle of 90 ◦ , as schematically depicted in Fig. 5.36 [18, 23, 24, 25]. According to Cogswell, the elongational rate and the elongational viscosity can be estimated by using the following equations:
5.3 Rheometry
169
Pressure transducer Recirculation or vortex zone Entrance pressure drop
Figure 5.36:
Schematic diagram of the Cogswell method
˙ =
2 4η γ˙ ap 3(n + 1)ΔPe
9(n + 1)2 ηe = 32η γ˙ ap =
4Q πR3
ΔPe γ˙ ap
(5.52) (5.53)
(5.54)
where ˙ is the elongational rate, η e is the elongational viscosity, η is the viscosity, γ˙ ap is the power law index, ΔP e is the pressure drop at capillary inlet (obtained from Bagley correction extrapolation at L/D = 0) , and n is the apparent shear rate – Binding’s method – This method is based in the theory developed by Cogswell, but it is a more accurate method since Binding does not neglect the WeissenbergRabinowitsch correction. However, this leads to more complex calculations [26, 27]. To model the shear viscosity, and the elongational viscosity Binding arbitrarily assumes a Power law model. – Semihyperbolically converging die – In this technique, schematically depicted in Fig. 5.37, the polymer flows through a cylindrical, converging die whose semihyperbolic shape leads to a shear-free flow within the die, assuming wall slip conditions. From the analysis of the relevant flow equations in the die, the use of a numerical method (typically the finite element method (FEM)) and the use of a constitutive equation, the elongational viscosity can be measured [28].
170
5 Melt Rheology
Figure 5.37:
Schematic diagram of a semihyperbolically converging extensional rheometer
INDUSTRIAL APPLICATION 5.2
Molecular Weight Distribution Comparison Using Rheometry Characterization Here, we will perform an analysis of molecular weight distributions of three different polypropylenes using storage modulus, G , and loss modulus, G , measurements as a function of the frequency. These measurements are presented in Table 5.13. Table 5.13:
Dynamic modulus for three different polypropylenes
PP - A G (Pa)
G" (Pa)
Frequency (rad/s)
0.1000
585
1340
0.2150
1190
2130
Frequency (rad/s)
PP - B G (Pa)
G" (Pa)
Frequency (rad/s)
PP - C G (Pa)
0.0300
585
1340
0.0300
1170
2680
0.0646
1190
2130
0.0646
2380
4250
G’" (Pa)
0.4640
2200
3330
0.1390
2200
3330
0.1390
4410
6650
1
3900
5010
0.3000
3900
5010
0.3000
7790
10000
2.1500
6480
7340
0.6460
6480
7340
0.6460
13000
14700
4.6400
10200
10300
1.3900
10200
10300
1.3900
20500
20500
10
15600
14200
3
15600
14200
3
31100
28400
21.5000
23000
19400
6.4600
23000
19400
6.4600
45900
38700
46.4000 100
32900 45000
25900 33700
13.9000 30
32900 45000
25900 33700
13.9000 30
65700 90100
51700 67300
5.3 Rheometry
171
Using the crossover 5 of storage modulus G and loss modulus G , the molecular weight distribution (MWD) of the different polypropylenes was compared. Figure 5.38 illustrates the relation between the crossover of G and G and the molecular weight distribution. As shown in the figure, as the molecular weight increases, the crossover modulus, G c , shifts to a lower frequency; and as MWD narrows, the crossover modulus shifts to a higher value.
Figure 5.38:
Correlation between the molecular weight distribution and the crossover modulus
Figure 5.39 presents plots of the dynamic modulus presented in Table 5.13. The crossover modulus allows the comparison of the molecular weight distribution (MWD) of the three polypropylenes.
Figure 5.39:
Comparison of the crossover modulus for the three different polypropylenes
Conclusions: When comparing the crossover modulus for the three polypropylenes, the following conclusions could be obtained: 5 The
crossover point is where G /G" = 1.
172
5 Melt Rheology
– The molecular weight for the three polypropylenes is M W C > M W B > M W A, because the order of the crossover on the frequency axis was G cA > GcB > GcC . – The molecular weight distribution of polypropylene C was narrower than polypropylenes A and B, because the order of the crossover on the modulus axis was G cC > GcA = GcB . – Because polypropylene C exhibited the higher molecular weight and the narrower MWD, it will more easily result in flow instabilities and can exhibit higher elastic effects in the molten state. – The shear viscosity curve can also be used to interpret molecular weight distributions. The smaller Newtonian plateau and the more gradual pseudoplastic decrease of viscosity exhibited by a polymer reflect a wider molecular weight distribution. The higher zero viscosity (the viscosity value in the Newtonian plateau) exhibited by the polymer reflects a higher average molecular weight of the polymer melt. To illustrate the correlation, Fig. 5.40 compares the viscosity curves for two different polymers at the same temperature. When looking at the figure, one can deduce that polymer B has the wider MWD because of its narrow Newtonian plateau and its more gradual pseudoplastic decrease of viscosity.
Figure 5.40: Correlation between MWD and the viscosity curves for two polypropylene melts at the same temperature
INDUSTRIAL APPLICATION 5.3
Flow Instabilities Study in a Thermoplastic Polymer In this case study, a mass flow of 70 kg/h of polypropylene was extruded through a rectangular die of 50 cm width, 10 cm length and 0.2 cm height, at 220 ◦ C. The particular PP to be extruded had the following rheological information based on a Bird-Carreau-Yasuda model: A = 4254 (Pa· s), B = 0.22 (s) , C = 0.63 , Tref = 243 (◦ C) , and U = 46015 (J/mol). The density of polypropylene at 220 ◦ C is 0.75 g/cm3 . The estimated pressure drop and the prediction of flow instabilities at the exit of the die were required under the specified conditions.
5.3 Rheometry
173
The volumetric flow was calculated as follows V =
m ˙ 70 · kg · h · 1000 · g · cm3 cm3 = = 25.93 . ρ h · 3600 · s · kg · 0.75 · g s
(5.55)
The shear rate for a non-Newtonian polymer was calculated with the following approximated equation: 6·V 0.772 · 6 · 25.93 · cm3 = = 60.04 · s-1 . 2 W ·h s · 50 · cm · 0.22 · cm2 The shear viscosity can be calculated using γ˙ =
1 U 1 ( − ) aT = exp R T Tref 1 1 46015 · J · mol · K ( − ) = exp mol · 8.3141 · J (220+273.15)K (243+273.15)K
(5.56)
(5.57)
= 1.65 η=
A · aT 4254Pa · s · 1.65 = = 979 · Pa · s. 0.63 (1+B · aT γ) ˙ c (1+0.22 · s · 1.65 · 64.04 s )
(5.58)
The pressure drop can be estimated using
ΔP =
12 · V · η · L 12 · 25.93 · cm3 · 979 · Pa · s · 10 · cm = = 7614111 · Pa W · h3 s · 50 · cm · (0.2)3 · cm3
7614111 · Pa · bar = 76.14 · bar. 105 Pa The shear stress can now be computed using ΔP =
(5.59)
(5.60)
MPa 60.04 · 6 = 0.06 · MPa. (5.61) s 10 · Pa Conclusions: An estimated pressure drop of 76.14 bar was predicted for the particular die and the given extrusion conditions. Because the calculated shear stress was below the critical shear stress for flow instabilities of polypropylene (0.13 MPa, see Table 5.1), flow instabilities were not predicted under the conditions stated in this case. τ = η · γ˙ = 979 · Pa · s
INDUSTRIAL APPLICATION 5.4
Modeling the Shear Viscosity Curves of a Polypropylene Here, a Bird-Carreau-Yasuda model was used to fit the data for a polypropylene polymer melt.
174
5 Melt Rheology
Table 5.14:
Capillary rheometry data for a polypropylene
Temperature, (◦ C) shear rate, (s-1 )
200 Viscosity, (Pa· s)
210 Viscosity,(Pa· s)
220 Viscosity, (Pa· s)
10
591
501
429 420
20
575
490
50
532
458
396
70
508
440
382
100
475
415
363
200
395
351
313
500
270
248
227
700
227
210
194
1000
185
172
161
2000
120
113
106
5000
64
61
58
7000
50
48
46
10000
39
37
35
The curve was obtained using a capillary viscometer, and the data was corrected according to the Ryder-Bagley and Weissenberg-Rabinowitsch correction. The data are presented in Table 5.14.
– The parameters of the Bird-Carreau-Yasuda model were obtained by fitting the experimental viscosity values and the calculated viscosities. Since polypropylene is a semicrystalline polymer, the best model to correlate the temperature dependence of viscosity is the Arrhenius model 1 U 1 aT = exp ( − ) (5.62) R T Tref η=
A · aT (1+B · aT γ) ˙ c
(5.63)
– The values of A, B, and C were obtained by minimization of the error between experimental data and the calculated data for the different shear rates and temperatures, according to the following error metrics Error =
M
(η calc − η exptal )
(5.64)
i=1
– When fitting the parameters of the Bird-Carreau-Yasuda model with the experimental data, a non-linear optimization was done using a commercial equation solving software.
5.3 Rheometry
175
Because of the non-linearity of the model, the initial guess values were very important for the convergence of the optimization. It was necessary to impose some limits to the A, B, and C values, as follows: • A is greater than zero • B is greater than zero • C is within the range of 0 to 1 (typically between 0.3 and 0.7) It was also important to limit the activation energy, U , to be greater than zero. Table 5.15 presents the optimized parameters of the Bird-Carreau-Yasuda model for the given polypropylene data. Figure 5.41 shows a comparison between the experimental data and the model.
Figure 5.41: Experimental data (symbols) and the model prediction (lines) for the viscosity of a polypropylene polymer melt
Table 5.15:
Parameters of the Bird-Carreau-Yasuda model for a polypropylene Parameter
Value
A, (Pa · s)
608.758667
B,(s) C
0.003946 0.744911
U , (J/mol)
32052.3943
Tref , (◦ C)
200
Conclusions: The obtained parameters of the Bird-Carreau-Yasuda model reproduced with good precision the experimental viscosity data for all three temperatures.
176
5 Melt Rheology
INDUSTRIAL APPLICATION 5.5
Regression Analysis of Rheological Data to Obtain the Phan-Thien and Tanner Multimode Model Parameters The parameters of the Phan-Thien and Tanner multimode model for a particular polypropylene to be used for fiber spinning applications were obtained by fitting the following rheological data: – Material – Homopolymer polypropylene with a MFI 16 g/10 m at 230 ◦ C/2.16 kg – Shear viscosity and Cogswell’s elongational viscosity – These data were measured in a capillary viscometer using a 1 mm circular capillary with 10, 20, 30, and 40 mm lengths. – Elongational viscosity – Measured at low elongational rates (in the Newtonian range, below 0.1 s-1 ) and computed using the Trouton viscosity equation. This relationship states that the elongational viscosity at very low elongational rates is equal to 3 times the shear viscosity. – Shear viscosity and complex modulus – These data was measured in a rotational rheometer with a 25 mm diameter plate-plate, 1 mm gap, 1% deformation (in the linear viscoelasticity range), and frequency within the range from 0.005 and 100 rad/s. The rheological data are presented in Fig. 5.42. The unfilled circles in the figure correspond to the shear viscosity obtained by capillary rheometry, and the filled circles correspond to the shear viscosity obtained by rotational rheometry and by applying the Cox-Merz principle of complex viscosity. The unfilled triangles in the figure correspond to loss modulus G obtained by rotational rheometry; the unfilled squares correspond to storage modulus G obtained by rotational rheometry; the unfilled rhomboids correspond to the elongational viscosity estimated by the Cogswell’s method; and the filled rhomboids correspond to the elongational viscosity estimated with the Trouton relationship. The parameters of the model were obtained by fitting the rheological data, according to the following procedure: – The values of λ i and Gi were obtained by fitting the storage and loss modulus, according to the equations
G =
N Gi (λi ω)2 i=1
1+(λi ω)2
N Gi λi ω G = 2 1+(λ i ω) i=1
(5.65)
5.3 Rheometry
Figure 5.42:
177
Dynamic test in a rotational rheometer [23, 24]
and the restriction ηo =
N
(5.66)
Gi λi .
i=1
– The values of ξ i were obtained by fitting shear viscosity according to η=
N i=1
Gi λi . 1+ξi (2 − ξi )(λi ω)2
(5.67)
– The λi , Gi and ξi values were obtained by minimization of the error between experimental data and the calculated data using the previous equations for the different frequencies, using the error metrics
Error =
M
Gicalc
2
M
−1 + Giexptal i=1 2 M ∗calc η + −1 . ∗exptal η i=1 i=1
2
Gicalc
Gi exptal
−1 (5.68)
– The values of i were obtained by fitting the Trouton viscosity at low elongational rates and the Cogswell’s elongational viscosity. To fit the elongational viscosity, it was necessary to solve by iteration the non-linear equation resulting from applying the Phan-Thien and Tanner equations to an elongational, uniaxial, and uniform flow at steady state:
178
5 Melt Rheology
i exp (τzzi +2τrri ) τzzi − 2λi (1 − ξi )τzzi ˙ = 2λi Gi , ˙ Gi
i (τzzi +2τrri ) τrri − 2λi (1 − ξi )τrri ˙ = 2λi Gi , ˙ exp Gi τzz =
N
τzzi
τrr =
i=1
N
η=
τrri
i=1
τzz − τrr .
˙
(5.69) (5.70)
(5.71)
– The ξi values were obtained by minimization of the error between experimental data and the calculated data using the above equations for the different frequencies, according to the error metrics Error =
M ηecalc i=1
Table 5.16: 230 ◦ C)
2
ηeexptal
−1
(5.72)
.
Parameters for Phan-Thien and Tanner model (PP PROPILCO 18H86 at
i
λi
i
ξi
Gi
1
2.51E-04
1.00E+00
8.60E-02
1.22E+05
2
1.66E-03
1.00E+00
3.06E-01
4.02E+04
3
5.94E-03
8.00E-01
9.43E-01
1.53E+04
4
2.07E-02
2.00E-01
5.89E-01
6.39E+03
5
9.00E-02
3.50E-02
8.30E-02
2.88E+03
6
4.86E-01
3.00E-02
2.62E-01
3.62E+02
7
2.71E+00
3.00E-02
9.99E-01
3.44E+01
When fitting the parameters of the Phan-Thien and Tanner multimode model with the experimental data, a non-linear optimization was done using a commercial equationsolving software. Table 5.16 shows the parameters of the Phan-Thien and Tanner multimode model for the given polypropylene. The solid lines of Fig. 5.42 correspond to the Phan-Thien and Tanner multimode model predictions. Conclusions: The Phan-Thien and Tanner multimode model agreed very well with the given experimental data. Small oscillations are visible because of the discretization of the relaxation spectra.
5.3 Rheometry
179
INDUSTRIAL APPLICATION 5.6
Modeling the Shear Viscosity Curves and Their Application in Injection Molding In this case study, the cold runners and gates of a six-cavity mold presented in Fig. 5.43 needed to be rheologically balanced, such that every cavity filled at the same time. The polymer melt data is given by, – Material – Injection molding grade polyamide 6 – Rheological data – The rheological properties were modeled at the injection temperature using the Bird-Carreau-Yasuda model with parameters, A = 373 (Pa· s), B = 0.12 s and C = 0.35. The details of the regression procedure to obtain the Bird-CarreauYasuda model parameters were presented in in a case study above. – Processing conditions and properties – The following injection molding conditions were set: part weight 20 g, total length of channel 250 mm, density at melt temperature 1.1 g/cm3 , injection molding speed 100 mm/s and thickness 2.5 mm. – Geometry of gates and runners – The dimensions of the runners and gate system are presented in Table 5.17. The diameters of channels 1 and 3 were used for balancing the runners and gate system. The values presented are the final values obtained after the iteration process. Table 5.17:
Dimensions of the runners and gate system of the six-cavity mold Diameter (mm)
Length (mm)
Number of channels
Channel 1
5.11
100
2
Channel 2
7
150
2
Channel 3
4.88
100
4
Gate I
1
1
2
Gate II
2
1
4
The first step is to calculate the total volume of the part, runners, and gates of the given mold system. The total volume was calculated by adding the volume of the six parts (calculated with the weight of the part and the melt density), the volume of the different channels and volume of the different types of runners (calculated with the given geometry),
VT ot =
N Wpart
i
i=1
ρmelt
− 1+
M j=1
Vchannelj +
P k=1
Vgatesk .
(5.73)
180
5 Melt Rheology
Figure 5.43:
Six-cavity mold to be rheologically balanced
The injection time was calculated by dividing the injection molding speed by the channel length, Vinj . (5.74) L The volumetric flow was calculated by dividing total volume by the injection time, t=
VT ot . V˙ = t Table 5.18: mold
(5.75)
Calculations for balancing the runners and gate system of the six-cavity
Diameter (mm)
Length (mm)
# of Channels
Flow rate (cm3 /s)
Shear Rate (1/s)
Viscosity (Pa· s)
Pressure drop (Bar)
Channel 1
5.11
100
2
8.8
547.4
85.8
45.1
Channel 2
7
150
2
17.6
426.7
93.4
41.9
Channel 3
4.88
100
4
8.8
629.3
81.7
51.7
Gate I
1
1
2
8.8
73186.6
15.5
55.8
Gate II
2
1
4
8.8
9148.3
32.2
7.2
Total
52.8
The volumetric flow of channels 1 and 3 was calculated as the total volume divided by 6. The volumetric flow of channels 2 was calculated as the total volume divided by 3. The shear rate for each channel and gate was approximated using 4 · V˙ . (5.76) π · R3 The viscosity was calculated with the Bird-Carreau-Yasuda model, and the pressure drop for each channel and gate was obtained with the expression γ˙ = 0.815 ·
η= ΔP =
A , (1+B · γ) ˙ c 8 · V˙ · η · L . π · R4
(5.77) (5.78)
5.3 Rheometry
181
Using an iterative scheme with the diameters of channels 1 and 3 and with the diameter of channel 2 fixed, the runners and gate system were obtained. Because the runners and gate system were balanced, the pressure drop through every channel was the same and equal to 100.9 bar. The maximum recommended shear rates for gates are presented in Table 5.19. For a PA6 a maximum shear rate of 60000 1/s is recommended. According to the calculations, the higher shear rate 73186.6 1/s was obtained. Table 5.19:
Maximum shear stress and shear rates for various polymers [29, 30] Polymer
Max. shear stress, (Pa)
Max. shear rate, (1/s)
PP
250,000
100,000
HDPE
200,000
40,000
LDPE
100,000
40,000
Flexible PVC
150,000
20,000
Rigid PVC
200,000
20,000
PS
250,000
40,000
HIPS
300,000
40,000
SAN
300,000
40,000
ABS
300,000
50,000
PA6
500,000
60,000
PA66
500,000
60,000
PET
500,000
6,000
PBT
400,000
50,000
PC
500,000
40,000
PMMA
400,000
40,000
PPS
345,000
50,000
PSU
500,000
50,000
PUR
250,000
40,000
Conclusions: Although the cold runners and gates system of the six-cavity mold was balanced, a shear rate at gate 1 exceeded the maximum recommended value; hence, a redesign of this gate must be done. INDUSTRIAL APPLICATION 5.7
Dispersion of a Polymer Blend Using a Single-Screw Extruder In this case study a 50% PP and 50% LDPE polymer blend was supposed to be processed in a 45-mm single-screw extruder. It was necessary to predict beforehand which polymer could be used as the dispersed phase to guarantee the required product
182
5 Melt Rheology
quality. The intended operating conditions of the extruder and the barrier gap geometry of the screw in the metering zone were: Temperature of the melt is 210 ◦ C Screw rotational speed, N = 45 rpm = 0.8 rev/s Barrier gap, δ = 0.3 mm The first step is to estimate the viscosities of both polymeric materials at the extrusion shear rate (γ) ˙ within by the 45-mm screw. The shear rate was calculated by the expression γ˙ =
π·D·N π · 45 mm · 0.8rev/s = = 377s-1 . δ 0.3 mm
(5.79)
Figures 5.44 and 5.45 present the viscosity curves of LDPE and PP as a function of shear rate and temperature of the melt.
Figure 5.44:
Viscosity curve of LDPE
The viscosity values of the polymer melts shown in Eq. 5.80 can be obtained from the curves in Fig. 5.44 and 5.45.
Figure 5.45:
Viscosity curve of PP
5.3 References
Figure 5.46:
183
Grace diagram [1]
ηLDP E = 80(Pa · s) ηP P = 250(Pa · s)
(5.80)
The viscosity ratios were also calculated to predict the dispersed phase reading in the Grace diagram and are shown in Eq. 5.81. ηLDP E 80 = 0.32 = ηP P 250 ηP P 250 = 3.12 = ηLDP E 80
(5.81)
From the Grace diagram in Fig. 5.46, it was clear that PP cannot be employed in the dispersed phase because the viscosity ratio η P P over ηLDP E was near the 3.8 limit. Because the ratio ηLDP E over ηP P was the lowest, for LDPE it is likely to be selected for the dispersed phase. Conclusion: The predicted dispersed phase for the physical blend of 50% PP and 50% LDPE at the given operating conditions was LDPE, and the matrix or continuous phase was PP.
References 1. F.T. Trouton. Proc. Roy. Soc. A, 77, 1906. 2. H. Muenstedt. Rheol. Acta, 14:1077, 1975. 3. C.D. Han and K.W. Len. J. Appl. Polym. Sci., 29:1879, 1984. 4. J.M. Castro and C.W. Macosko. AIChE J., 28:250, 1982. 5. J.M. Castro, S.J. Perry, and C.W. Macosko. Polym. Comm., 25:82, 1984. 6. G.K. Batchelor. Annu. Rev. Fluid Mech., 6:227, 1974.
184
5 Melt Rheology
7. A. Einstein. Ann. Physik, 19:549, 1906. 8. P. Geisb¨usch. PhD thesis, IKV-RWTH-Aachen, Germany, 1980. 9. E. Guth and R. Simha. Kolloid-Zeitschrift, 74:266, 1936. 10. E. Guth. Phys. Rev., 53:321, 1938. 11. R.K. Gupta. Flow and rheology in polymer composites manufacturing. Elsevier, Amsterdam, 1994. 12. W.J. Milliken and R.L. Powell. Flow and rheology in polymer composites manufacturing. Elsevier, Amsterdam, 1994. 13. J.F. Agassant, P. Avenas, J.-Ph. Sergent, and P.J. Carreau. Polymer Processing - Principles and Modeling. Hanser Publishers, Munich, 1991. 14. J. P. Sergent J. F. Agassant, P. Avenas and P. J. Carreau. Polymer processing: principles and modelling. Hanser Publishers, Munich, 1991. 15. R. I. Tanner, N. Phan-Thien. J. Now-Newt Fluid Mech, 2:353, 1977. 16. D. Acierno, F. P. La Mantia, G. Marrucci, and G. Titomanlio. J. Now-Newt Fluid Mech, 1:125 –147, 1977. 17. H. A. Barnes and D. Bell. Controlled-stress rotational rheometry: A historical review. KoreaAustralia Rheol. J, 15(4):285–336, 2003. 18. C.W. Macosko. RIM Fundamentals of reaction injection molding. Hanser Publishers, Munich, 1989. 19. B. N. Wang, M. Sentmanat. and G. H. McKinley. Measuring the transient extensional rheology of a LDPE melt using the SER universal testing platform. J. Rheol, pages 1–29, 2004. 20. M. Sentmanat. Miniature universal testing platform: from extensional melt rheology to solid-state deformation behavior. Rheol Acta, 43:657–669, 2004. 21. RHEOTENS 71.97. The New Tensile Tester for Polymer Melts. G¨ottfert, Germany. 22. A. Bernnat. Polymer Melt Rheology and the Rheotens Test. PhD thesis, Institut f¨ur Kunststoffverarbeitung, 2001. 23. J. Sierra. Development of biconstituent fibers of polypropylene and polyamide 6 for high speed fiber spinning. PhD thesis, University of the Vasc Country, Spain, 2005. 24. M. P. Noriega, J. D. Sierra, I. D. L´opez, and I. Katime. Biconstituent fibers from polypropylene and polyamide 6: fiber spinning modeling and properties. WAK Kunststoffetechnik, 2:1–28, 2006. 25. F. N. Cogswell. Polymer melt rheology: A guide for industrial practice. John Wiley & Sons, 1981. 26. D. Binding. An approximate analysis for contraction and converging flows. J. Non-Newtonian Fluid Mech, 27:173–189, 1988. 27. M. Gupta. Effect of elongational viscosity on axisymmetric entrance flow of polymers. Polym. Eng. Sci., 40, 23, 2000. 28. K. Feigl, F.X. Tanner, B.J. Edwards, and J.R. Collier. A numerical study of the measurement of elongational viscosity of polymeric fluids in a semihyperbolically converging die. To appear in Journal of Non-Newtonian Fluid Mechanics. 29. H. Rees. Mold Engineering. 2nd edition Hanser Publishers, Munich, 2002. 30. J. P. Beaumont. Runner and Gating Design Handbook: Tools for Successful Injection Molding. 2nd edition Hanser Publishers, Munich, 2007. 31. T. A. Osswald and G. Menges. Material Science of Polymers for Engineers. Hanser Publishers, Munich, 2nd edition, 2003.