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Dependence of Elastic Quantities on Experimental Parameters
5
Dependence of Elastic Quantities on Experimental Parameters
5.1 Recoverable Compliance Creep compliances and recoverable compliances are usually measured as functions of time. In the linear range of deformation, retardation spectra may be determined from the time dependence of the recoverable compliance. As it becomes and the recoverable compliance obvious from Figure 4.3, the creep compliance generally depend on the stress applied, whereby reacts to the stress less . sensitively than
5.1.1 Stress Dependence In Figure 5.1 the stress dependence of the steady-state recoverable compliances and of the steady-state viscosities , calculated from the creep compliances according to Equation 3.3, is quantified for a commercial polypropylene that possesses a linear molecular structure. For an easier comparison, the steady-state elastic compliance is normalized by its linear value and the viscosity by the zero-shear viscosity . Recoverable compliance and viscosity exhibit a linear stress-independent range that is distinctly wider for the viscosity than for the elastic compliance. Furthermore, the elastic compliance shows a much stronger decay with shear stress than the viscosity.
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5 Dependence of Elastic Quantities on Experimental Parameters
1.4
180°C
1.2 Je/J0e 1
1.4 1.2 1
0.8
0.8
0.6
0.6
0.4
0.4
norm Je, norm
0.2 1
0.2 10
100
1000
[Pa]
10000
Figure 5.1 Normalized stationary viscosities and normalized stationary elastic as functions of constant creep stresses for a commercial polypropylene compliances at 180 °C. is the linear steady-state recoverable compliance and the zero-shear viscosity. Adapted from [5.18]
5.1.2 Temperature Dependence Another important experimental parameter is the temperature. Figure 5.2 presents the creep compliances and the recoverable compliances at = 10 Pa and temperatures of 180, 200, and 220 °C for a commercial linear polypropylene. The experiments have been conducted in the linear range of deformation. The compliance, which is governed by the viscosity at longer times, exhibits a distinct dependincreases with growing temperaence on the temperature in the sense that ture, corresponding to a decreasing viscosity. However, the steady-state recoverable compliance is found to be independent of the temperature. The time-dependent regime of the recoverable compliance reflects a somewhat faster recovery at higher temperatures, as expected from the shorter retardation times due to the lower viscosities.
5.1 Recoverable Compliance
10-1
J(t), J r(tr) [Pa -1]
Polypropylene =10 Pa
10-2
J(t)
10-3
Jr(tr) 10-4
10-5 10-1
180 °C 200 °C 220 °C 100
101
102
103
t, tr [s]
104
Figure 5.2 Creep compliance as a function of creep time and recoverable compliance as a function of recovery time for a commercial polypropylene in the linear range at the temperatures 180, 200, and 220 °C. Reprinted with permission from [5.4]; copyright 2009 American Chemical Society
It is interesting to note that for the curves of and , measured at the different temperatures, master curves can be obtained by shifting them along the time (see Figure 5.3). This means that the examined axis with the same shift factor polypropylene shows a thermorheologically simple behavior. Further details on thermorheological properties of polymer melts can be found in [5.1], for example.
J(t/a T ), J r (t r /a T ) [Pa -1 ]
10-1
Polypropylene =10 Pa T0 =180 °C
10-2
10-3
10-4
180 °C 200 °C 220 °C 10-5 10-1
100
101
102
103
104
t/aT , t r /a T [s]
Figure 5.3 Master curves of and from Figure 5.2 at the reference temperature = 180 °C. Reprinted with permission from [5.4]; copyright 2009 American Chemical Society
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5 Dependence of Elastic Quantities on Experimental Parameters
The independence of the steady-state recoverable compliance from temperature is valid in the nonlinear regime too, as can be concluded from the measurements at a creep stress of 1000 Pa presented in Figure 5.4 for a commercial polypropylene. 10-3 = 1000 Pa
T [°C]
-1
Jr [Pa ]
180 200 220
10-4
PP 10 10
-5
10-1
100
101
102
103
t r [s]
104
Figure 5.4 Recoverable compliance as a function of the recovery time for a commercial polypropylene at different temperatures and a creep stress of 1000 Pa in the nonlinear regime. Reprinted with permission from [5.4]; copyright 2009 American Chemical Society
A steady-state recoverable compliance independent of temperature is reported in [5.2] for a linear low density polyethylene, too, and in [5.3] for polystyrene and polystyrene blends, for example. The existence of master curves for the time dependence of the recoverable compliance is not a general feature of polymer melts, as it becomes obvious from Figis presented at various temperatures for a long-chain branched ure 5.5, where low density polyethylene (LDPE) [5.4]. For a clearer distinction the compliance is plotted on a linear scale, while for the time the logarithmic scale is kept. Evidently, the steady-state values of the recoverable compliances decrease with growing temperatures and consequently a master curve cannot be obtained by a mere shift along the time axis. Such a behavior is found for some other polymer melts too. The temperature dependencies of various polymer melts are discussed in Section 6.4.1.3 in relation to their molecular structure.
5.2 Relaxation Modulus
1.6 1.4
Jr
T 130°C 150°C 170°C 190°C
1.0
-3
-1
Jr(tr) [10 Pa ]
1.2
0.8 0.6 0.4 0.2 0.0 -1 10
LDPE-tub 4 10
0
10
1
2
10 t r [s]
10
3
10
4
10
5
Figure 5.5 Recoverable compliance as a function of the recovery time in a semi-logarithmic plot for a long-chain branched LDPE at four temperatures in the linear regime. Reprinted with permission from [5.4]; copyright 2009 American Chemical Society
5.2 Relaxation Modulus The relaxation modulus of polymer melts is distinctly time-dependent. From the time dependence in the linear range the relaxation spectrum can be determined according to Equation 3.18 or, in the discrete form, from Equation 3.16. Measurements of the relaxation modulus are subject to restrictions with respect to the time window and, subsequently, to the determination of relaxation times at their lower and higher ends. The shortest measurable times are given by the duration of the generation of the deformation step, and the longest times depend on the resolution and noise level of the torque or force transducer. Obviously, the limits are specific of the technical performance of the rheometer used. As an example, in [5.5] a time window between about 0.1 and 30 s is reported for a standard polystyrene characterized with a usual laboratory rotational rheometer. In practice, the relaxation spectra are more often determined from dynamic-mechanical experiments as described in Section 3.3, because the underlying experimental technique and the numerical procedures have been developed to a high standard. Relaxation experiments in the nonlinear regime have gained a special importance, however, to determine the so-called damping function, which makes possible a separation of the time and deformation dependence of the modulus. A review on
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5 Dependence of Elastic Quantities on Experimental Parameters
damping functions and their applications in constitutive equations is given in [5.6]. The basic concept of time-deformation separability becomes obvious from Figure 5.6. The modulus curves covering about three decades as functions of time decrease with growing deformation steps . But they exhibit very similar shapes as functions of the duration of the relaxation experiments and can be superposed by shifting them along the axis of the modulus on the logarithmic scale. The shift .Taking factors for the different shear steps constitute the damping function , then the relation the curve at the smallest step as the linear reference (5.1) is valid. This experimental finding means that the relaxation modulus dependent divided by the damping function on time and shear can be obtained from describing the nonlinearity.
G(t, ) G(t, )/h( )
42
PS-r-95 T = 200 °C
10 6 Pa 10 5 10 4
G(t, ) / h( )
10 3
x 10
G(t, ) 10 2 10 1
deformation (from top to bottom): 0.1 1.0/1.5/2.0/2.5/3.0/3.5/4.0 4.5 0.1
0.5
1
2 t [s]
Figure 5.6 Relaxation experiments on a standard polystyrene at a temperature of 200 °C and various deformation steps between 0.1 (dotted line) and 4.5 (broken line). represents a master curve obtained by shifting the various modulus curves along the modulus describes the shift factors. For the matter of axis to the curve at a step of = 0.1 [5.7]. convenience and to point out the functionality, the strain steps are designated by instead of and for a clearer presentation is multiplied by 10
For the polystyrene of Figure 5.6 the damping function is plotted in Figure 5.7. At = 1 can be assumed approxsmall steps up to about 0.2 a linear behavior with imately. For higher deformation steps the damping function goes down, but within the accuracy of the measurements it is the same at the two different temperatures 180 and 200 °C. The damping function is not a universal relation and depends on
5.2 Relaxation Modulus
the type of polymer and its molecular structure, as can be seen from [5.7] for polystyrenes and from [5.8] for polyolefins.
h( )
Eq. (5.2) with = 7.8
1
0.1
PS-r-95 (T = 180 °C) PS-r-95 (T = 200 °C) 0
1
2
3
4
5
Figure 5.7 Damping function versus the deformation step for the polystyrene of F igure 5.6 at two temperatures [5.7]. The full line is drawn according to Equation 5.2 with the parameter = 7.8
A general numerical description of the experimental curves based on a molecular theory has not been achieved. Rather, various empirical formulas can be found in the literature, which are reviewed in [5.6]. The full line in Figure 5.7, for example, represents the so-called PSM function (5.2) named after the three persons who proposed it [5.9]. is a fit parameter and was found to be 7.8 for the measurements in Figure 5.7 [5.7]. For some polymers the damping function can be described numerically by the sum of two exponential functions, as shown in [5.10]: (5.3) , , and are material-specific parameters obtained from a numerical description of the experimental damping function with Equation 5.3. The advantage of this exponential damping function is that it can be handled mathematically in an easy way.
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5 Dependence of Elastic Quantities on Experimental Parameters
5.3 Storage Modulus The storage modulus measured in the linear viscoelastic regime may change over several decades in the frequency range applicable for commercial rheometers. An example is given for six anionic polystyrenes with very narrow molar mass distribetween 1.0 and 1.1 in butions characterized by polydispersity indices describes the number average molar mass, and the weight average Figure 5.8. of the samples range between 125 and 750 kg/mol. molar masses
Figure 5.8 Storage modulus as a function of the angular frequency at 180 °C for six anionic polystyrenes with very narrow molar mass distributions and weight average molar masses ranging between 125 and 750 kg/mol from bottom to top
The importance of a wide frequency range becomes obvious from Figure 5.8. In the double-logarithmic plot at low frequencies the terminal regime with the slope of 2 is very pronounced and the linear steady-state quantities and can be determined when making use of the corresponding plot of the loss modulus and the Equations 3.30 and 3.32. At higher frequencies, is only very weakly dependent on frequency, before it starts to rise again. This range is called the rubber-elastic plateau, because a frequency-independent storage modulus is typical of crosslinked rubber (see, e. g., [5.1]). For thermoplastic polymers the rubber-elastic behavior is due to entanglement networks of molecules (see Chapter 7). According to the theory of rubber elasticity, for the plateau modulus, designated as , the relation (5.4)
5.4 Normal Stress Difference
is valid (see [5.19], for example). is the number of cross-links or entanglements per volume, the Boltzmann constant, the absolute temperature, the polymer the molar mass of a molecule strand between two cross-links or entandensity, glements, and the universal gas constant. This relation is the basis for the experimentally determined entanglement molar masses published in [5.20], for example. In [5.1] the relation (5.5) is presented. The factor 2 goes back to the assumption of a fourfold functionality of entanglements. It has to be mentioned, however, that curves of like in Figure 5.8 can be found for narrowly distributed polymers only. For materials with wider molar mass distributions the terminal regime often cannot be reached in the low frequency range attainable and the plateau is often smeared out in such a way that it is difficult to be determined. As it is well known and can be found in many text books on polymer rheology, the temperature dependence of properties of a great number of polymer melts is thermorheologically simple. That means that the modulus curves at different temperatures can be shifted along the logarithmic frequency axis in good approximation to a master curve, while the shift along the modulus axis is negligible. On the other at hand, the existence of such a master curve makes it possible to obtain various temperatures by measuring the frequency dependence at one temperature and shifting this curve with the shift factors known.
5.4 Normal Stress Difference As shown in Figure 4.6, the normal stress difference is a distinct function of time and shear rate. The measurements give a hint that a time-independent value may not be easy to attain. One reason for these difficulties is a fractioning of the polymer melt at the rim of the rheometer plate, particularly at higher shear. With special arrays and experimental care, measurements of steady-state normal stress differences have become possible and are reported in the literature. An example is given in Figure 5.9 for LDPE presenting the normal stress coefficient together with the viscosity [5.10]. The experiments were performed over a wide range of shear rates at the various temperatures indicated in the figure. A master curve for the viscosity function was obtained in the usual way by multiplying
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5 Dependence of Elastic Quantities on Experimental Parameters
the shear rate with the shift factor aT and dividing the viscosity by it. Due to the scaling of n1 with the reciprocal shear rate squared according to Equation 3.39, . The experimentally obthis quantity has consequently been reduced with tained master curve does exhibit a small scatter only and thus represents the normal stress coefficient n1 over a wide range of shear rates or shear stresses, respectively. A constant n1 marking the linear behavior is only found in a narrow range of shear rate. From this plateau, n1 decreases very strongly over five decades as a function of shear rates ranging from 10−3 to 102 s−1. This decay is distinctly more pronounced than that of the viscosity. Similar to the stress dependencies of the viscosity and the recoverable compliance in Figure 5.1, the results of Figure 5.9 confirm a stronger nonlinearity of the elastic in comparison with the viscous behavior. The results of Figure 5.9 show that normal stress coefficients at various temperatures can be derived from the master curve of a thermorheologically simple polymer melt. However, for the stress or rate dependence of n1 a description by a model is not available yet.
Figure 5.9 Steady state values of the viscosity h and the normal stress coefficient n1 obtained from stressing experiments for LDPE. The measurements were performed at the different temperatures indicated and shifted to master curves by using the shift factor aT. The reference temperature was T0 = 150 °C. Adapted from [5.10]
An interesting empirical relation between and reading moduli
and the frequency-dependent
(5.6) is reported in [5.11]. For a typical LDPE it is shown in Figure 5.10 that the validity of this relation is fulfilled with a surprisingly high accuracy, when the shear rate of
5.4 Normal Stress Difference
the stressing experiment is set equal to the angular frequency of the dynamic- mechanical experiment. Equation 5.6 is interesting insofar as it correlates measurements of normal stresses in the nonlinear regime with those of the dynamic moduli in the linear range. Thus, it can be considered as an analog to the Cox-Merz relation that describes the equality of the shear viscosity as a function of the shear rate and the dependence of the magnitude of the complex viscosity on the angular frequency, when shear rate and frequency are chosen the same (see [5.1], for example).
Figure 5.10 Comparison of the first normal stress coefficient as a function of the shear rate with the quantity as a function of the angular frequency , derived from dynamic- mechanical measurements according to Equation 5.6 for LDPE. For an easier comparison, the measured values of the normal stress coefficients are presented as a continuous curve. Adapted from [5.11]
In [5.11] it has been shown that Equation 5.6 is valid for a polystyrene, a high density polyethylene, and a polypropylene too. Because of the purely empirical nature of Equation 5.6, its validity has to be proven for each material separately. Normal stress measurements are often overinterpreted. For example, sometimes Equation 3.41, exactly valid in the linear regime, is applied to determine the recovin the nonlinear range from the first normal stress difference , erable shear according to (5.7) with s being the shear stress. As can be expected, however, this equation may be considered as a rough approximation only (cf. Table 12.1).
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5 Dependence of Elastic Quantities on Experimental Parameters
5.5 Recoverable Elongation As described in Section 4.2.6, the recovery of an elongated polymer sample is a function of time. This time dependence may superpose in a complex way the influences of external parameters like total strain, stress, or temperature. Therefore, in the following, equilibrium data are discussed. In Figure 5.11 the dependence of the steady-state recoverable strain on the total strain is presented for LDPE at three elongational rates and a temperature of 150 °C. It is evident that the recoverable strain significantly depends on the total strain and the strain rate.
Figure 5.11 Steady-state recoverable strain as a function of the total strain for the LDPE IUPAC A at T = 150 °C and various elongational rates. Each symbol marks a separate sample that was stretched up to the different elongations and then recovered by cutting it into pieces according to the method sketched in Section 4.2.6. Adapted from [5.17]
For all three elongational rates chosen, increases with and reaches a plateau value that becomes higher the larger the elongational rate . The broken line in Figure 5.11 marks the case for a total reversibility of the elongation applied. At high elongational rates and rather small strains or short times, respectively, most of the deformation is reversible, but at higher elongations the deformation contains a significant viscous portion. In Figure 5.12 the recoverable strains for polystyrene as a function of the tensile stress at 130 °C are presented. describes the recoverable strain in the regime where it is independent of the total deformation. The recoverable strain increases
5.5 Recoverable Elongation
with the stress applied and then starts to level off. This behavior indicates that the recoverable part of the elongation approaches a limiting value that can be assumed to be reached when all molecules are stretched. The steady-state recoverable compliance defined as (5.8) is presented in Figure 5.12 too. This material function is constant at small stresses, designating the linear range of deformation, and decreases at higher stresses, marking the nonlinear regime.
rs
10-4
Polystyrene T=130°C
De
De [Pa-1]
0
10
10-5
10-1
rs
10-6
10-2
10-7
E=const. . =const. 10-3
103
104
105
106
E [Pa]
107
10-8
Figure 5.12 Steady-state recoverable strain and recoverable tensile compliance as functions of the tensile stress for polystyrene at 130 °C. The triangles mark results from experiments at constant stress, the circles those at constant strain rate. Adapted from [5.15]
It is worth noting that the steady-state values of the recoverable strain from experiments at constant stresses (creep experiments) and those at constant elongational rates (stressing experiments) come to lie on the same curve, when elongational rate is converted to tensile stress by making use of the elongational viscosity measured. This result is very reasonable, because the conformation of the molecules in an equilibrium state should not be dependent on the way in which it is attained. The agreement stands for the reliability of the two experimental modes. Another external parameter is the temperature. In Figure 5.13 the steady-state refor LDPE is shown in a temperature range between 125 and coverable strain 180 °C. The samples were elongated at various constant stresses [5.17]. Within the experimental accuracy the recoverable strain is independent of the temperatures chosen. This behavior is different from the significant temperature dependence of
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5 Dependence of Elastic Quantities on Experimental Parameters
the elongational viscosity, as shown in [5.17], for example. As seen from Figure 5.13, the recoverable strain of the LDPE increases with tensile stress, similar to the behavior of the PS presented in Figure 5.12.
Figure 5.13 Steady-state recoverable strain at various temperatures for LDPE. The previous elongations were performed at the indicated constant tensile stresses up to a stretching ratio of 40, corresponding to a Hencky strain of 3.7 at which a steady state of creep was achieved in all cases. Adapted from [5.17]
5.6 Extrudate Swell 5.6.1 General Features of Extrudate Swell The extrudate swell is a complex function of various external parameters. This fact is obvious from Figure 5.14, which comprehensively presents the various dependaccording to encies [5.1]. In this figure the extrudate swell, defined as of the capEquation 3.43, is plotted as a function of the length to radius ratio illary, for samples of a commercial polystyrene annealed in a silicone oil bath after extrusion. The general feature is that the extrudate swell decreases with the capillary length from comparatively high values obtained for very short capillaries and , which is smaller by a factor of about 2 seems to approach a plateau for large in comparison with an orifice. Similar results for the extrudate swell have been reported for other polymer melts too. In [5.12], for example, the extrudate swell of
5.6 Extrudate Swell
a high density polyethylene has been presented at different shear rates for capillarto . The extrudate swell increases with growies ranging from and approaches a plateau at larger capillary ing shear rate. It decreases with lengths. The existence of a significant extrudate swell for an orifice of very small length that can be deduced from Figure 5.14 indicates the role of the entrance flow for this quantity. As measured by laser Doppler velocimetry, large elongational components may occur in the entrance region of a capillary [5.13]. In the case of visco elastic materials, some portion of the deformation is elastic and recovers when the tensile stress becomes zero within the capillary. This recoil of the elongated molecules should be more pronounced the longer their way through the capillary, that is, the larger the capillary length at a given throughput.
Figure 5.14 Extrudate swell of a commercial polystyrene as defined by Equation of capillaries at two different stresses 3.43, as a function of the length to radius ratio at the wall and three extrusion temperatures . After extrusion, the samples were annealed in a silicone oil bath to reach their equilibrium. Adapted from [5.1]
It is a well-known feature, however, that there is still a remarkable extrudate swell measurable even for long capillaries, when the elastic deformation generated by the entrance flow should have recovered within the die. This effect goes back to the elastic portion of the shear in the capillary due to normal stresses. The release of the exiting strand from the geometrical constraints within the capillary leads to the recovery of the reversible deformation during shear and of the orientations left from the stretching of molecules in the entrance region of the capillary. The recovery of these two parts outside the capillary results in the observed extrudate swell.
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5 Dependence of Elastic Quantities on Experimental Parameters
From Figure 5.14 it is evident that within the accuracy of the measurements the . This means that, for the polyextrudate swell does not change much for styrene studied, the elastic deformation exerted by the elongational deformation in the entrance region of the die used has been recovered to a great deal by passing the die, and the extrudate swell is mainly determined, then, by normal stress effects. At a given die geometry this recovery process depends, of course, on the retardation times of the melt and thus on molar mass and temperature. From Figure 5.14 it follows that the extrudate swell increases with the shear stress at the wall, determined according to Equation 4.4, and that it is independent of temperature within the range chosen. However, due to the complexity of extrudate swell, these apparently obvious results have to be discussed in some more detail.
5.6.2 Detailed Analysis of Extrudate Swell Rather easy to analyze qualitatively is the stress dependence of the extrudate swell for short capillary lengths where the elongational deformation exerted at the die entrance is dominant. A result of the deformation of a melt in the entrance region is the so-called entrance pressure loss. As shown in [5.1] for polystyrene and visualized in Figure 12.2, this quantity increases with the shear rate and, consequently, the shear stress at the wall. For linear polymers, the entrance pressure loss is primarily due to extensional stresses [5.14]. Therefore, higher extensional stresses in the entrance region of a capillary are related to higher shear stresses at the wall. The higher stresses go along with larger extensions in the entrance region that lead to an increase in the recoverable elongation , as demonstrated in Figure 5.12 and Figure 5.13, for example, and consequently to a larger extrudate swell in the case of short capillaries, for which recovery processes of the entrance flow within the die are negligible. The independence from temperature of the extrudate swell in Figure 5.14 corresponds to that of the recoverable strain of the at small LDPE in Figure 5.13 and that reported in [5.15] for two polystyrenes. For long capillaries, in which the elastic deformation exerted in the entrance region may totally recover, recoverable shear becomes the source of elasticity. Also in this case the results of the extrudate swell presented in Figure 5.14 remain in qualitative accordance with the known dependencies of elastic quantities on external parameters, because the recoverable shear increases with stress as can be deduced from Figure 5.1 and is independent of temperature (see Figure 5.4), similar to the corresponding quantities in elongational flow according to Figure 5.12 and Figure 5.13. More difficult to analyze is the extrudate swell for capillaries where the recovery of the elastic part of elongation and the elastic part of shear superimpose. Comparable measurements of the extrudate swell of different polymers are frequently per-
5.6 Extrudate Swell
formed with a given capillary at constant apparent shear rates. Characterizations of this kind are not very meaningful for the following reason: The constant apparent shear rates result in the same throughput times for a chosen capillary, of course, and, consequently, comparable time windows for the viscoelastic effects. But different viscosities lead to different stresses. The consequence is that elastic properties changing with stress and their effect on the extrudate swell have to be taken into account. Additionally, the sample with the higher viscosity relaxes more slowly due to the inherent longer relaxation time. Choosing a constant stress at the capillary wall is the other experimental mode that can be performed. Then, for a given capillary, the apparent shear rates and throughput times change according to the viscosity of the sample. Consequently, the recovery within the capillary may be different and, following from that, the extrudate swell at the exit too. Assuming that for two samples the same elastic deformation is imposed at the capillary entrance, the material with the lower viscosity has the larger apparent shear rate and the shorter throughput time at comparable stresses. Therefore, even if the relaxation times of the samples were the same, the recovery within the capillary would be slower for the sample of low viscosity. But because of the relaxation time of the material becoming shorter with decreasing viscosity, the recovery is faster at comparable throughput times and the two effects may compensate each other to some extent. This discussion demonstrates the complexity regarding the detailed interpretation of extrudate swell measurements with respect to rheological properties.
5.6.3 Extrudate Swell for Various Die Geometries The discussion above makes it obvious that the shape of a die may have a significant effect on the extrudate swell. These dependencies are interesting for the assessment of the role of extrudate swell in cases of processing, as discussed in Sections 12.4 and 12.5. Some tendencies can be derived from results of laboratory experiments on LDPE reported in [5.16]. At a constant throughput, the extrudate swell of a tapered capillary with an entrance radius of 6 mm and an exit radius of 1 mm was compared with that of the same tapered die, but with a circular capillary of a radius of 1 mm and a length of 20 mm mounted at the exit. The extrudate swell measured by the ratio of the radius of the extruded strand to that of the capillary at its exit decreased by about 20 %. This finding is to be expected, due to the partial recoil of the molecules in the circular capillary that were previously stretched during the extensional flow in the tapered section. The extrudate swell could be further reduced by a trumpet-shaped outlet with a radius of 2 mm at the exit. This geometry enables a partial recoil of the molecules before the free total recovery after exiting the die resulting in the decreased extrudate swell. This example shows
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5 Dependence of Elastic Quantities on Experimental Parameters
how complex the assessment of the extrudate swell in a real extrusion process may become, because a melt often passes various geometries in a tool. Furthermore, the shear stresses within a real tool may change due to varying geometries. Moreover, another feature comes into play for an assessment of the effect of extrudate swell in processing. In real extrusion processes the equilibrium data presented in Figure 4.7 are seldom reached, because for practical reasons the extruded item is quickly cooled and subsequently the molecular motions are frozen when the melt exits the die. If a calibration step is used within an extrusion process, free expansion of a strand is not possible, and in the case of an applied pulling force a tension counteracts the extrudate swell (see Section 12.7). Two other factors have to be considered additionally and aggravate a quantitative description of extrudate swell even in a simple circular die. The shear stress or shear rate, respectively, are not constant across the cross section of a capillary and the elongational flow in the entrance region may be of a complex multiaxial nature. Regarding the stresses in Figure 5.14 at which capillary rheometry is usually undertaken, these are much higher than in the linear range of the creep recovery experiments presented in Figure 4.3, for example. Thus, it has to be assumed that all the extrudate swell experiments are performed in the nonlinear range of deformation. This feature makes a quantitative correlation with molecular data difficult, because melt properties change with stress and, thus, characteristic quantities are not available.
5.7 References [5.1]
Münstedt H., Schwarzl F. R., Deformation and Flow of Polymeric Materials, Springer, Berlin (2014)
[5.2]
Gabriel C., Münstedt H., Creep recovery behavior of metallocene linear low density polyethylenes, Rheol. Acta 38 (1999), 393–403
[5.3]
Orbon S. J., Plazek D. J., Recoverable compliance of a series of bimodal molecular blends of polystyrene, J. Polym. Sci. B 17 (1979), 1871–1890
[5.4]
Resch J. A., Stadler F. A., Kaschta J., Münstedt H., Temperature dependence of the linear steadystate compliance of linear and long-chain branched polyethylenes, Macromolecules 42 (2009), 5676–5683
[5.5]
Hepperle J., Münstedt H., Rheological properties of branched polystyrenes: nonlinear shear and extensional behavior, Rheol. Acta 45 (2006), 717–727
[5.6]
Rolon-Garrido V. H., Wagner M. H., The damping function in rheology, Rheol. Acta 48 (2009), 245–284
[5.7]
Hepperle J., Einfluss der molekularen Struktur auf rheologische Eigenschaften von Polystyrol- und Polycarbonatschmelzen (Influence of the molecular structure on rheological properties of poly styrene and polycarbonate melts), Doctoral Thesis, University Erlangen-Nuremberg, Shaker Verlag, Aachen (2002)
5.7 References
[5.8]
Stadler F. J., Auhl D., Münstedt H., Influence of the molecular structure of polyolefins on the damping function in shear, Macromolecules 41 (2008), 3720–3726
[5.9]
Papanastasiou A. C., Scriven L. E., Macosko C. W., An integral constitutive equation for mixed flows: Viscoelastic characterization, J. Rheol. 27 (1983), 387–410
[5.10] Laun H. M., Description of the non-linear shear behaviour of a low density polyethylene melt by means of an experimentally determined strain dependent memory function, Rheol. Acta 17 (1978), 1–15 [5.11]
Laun H. M., Prediction of elastic strains of polymer melts in shear and elongation, J. Rheol. 30 (1986), 459–501
[5.12] Han C. D., Charles M., Philippoff W., Rheological implications of exit pressure and die swell in steady capillary flow of polymer melts. I. Primary normal stress difference and the effect of L/D ratio on elastic properties, Trans. Soc. Rheol. 14 (1970), 393–408 [5.13] Wassner E., Schmidt M., Münstedt H., Entry flow of a low-density polyethylene melt into a slit die: An experimental study by laser-Doppler velocimetry, J. Rheol. 43 (1999), 1339–1353 [5.14] Münstedt H., Schwetz M., Heindl M., Schmidt M., Influence of molecular structure on secondary flow of polyolefins as investigated by laser-Doppler velocimetry, Rheol. Acta 40 (2001), 384–394 [5.15] Münstedt H., New universal extensional rheometer for polymer melts. Measurements on a polystyrene sample, J. Rheol. 23 (1979), 421–436 [5.16] Laun H. M., Schuch H., Transient elongational viscosities and drawability of polymer melts, J. Rheol. 33 (1989), 119–175 [5.17]
Münstedt H., Laun H. M., Elongational behaviour of a low density polyethylene melt. II. Transient behavior in constant stretching rate and tensile creep experiments. Comparison with shear data. Temperature dependence of the elongational properties, Rheol. Acta 18 (1979), 492–504
[5.18] Resch J. A., Kaschta J., Wolff F., Münstedt H., Influence of molecular parameters on the stress dependence of viscous and elastic properties of polypropylenes in shear, Rheol. Acta 50 (2011), 53–63 [5.19] Ferry J. D., Viscoelastic Properties of Polymers, John Wiley, New York (1980) [5.20] Fetters L. J., Lohse D. J., Colby R. H. in: Mark J. E. (ed.), Physical Properties of Polymers Handbook, Springer, Berlin (2005)
55
Dependence of Elastic Quantities on Experimental Parameters
5.1 Recoverable Compliance Creep compliances and recoverable compliances are usually measured as functions of time. In the linear range of deformation, retardation spectra may be determined from the time dependence of the recoverable compliance. As it becomes and the recoverable compliance obvious from Figure 4.3, the creep compliance generally depend on the stress applied, whereby reacts to the stress less . sensitively than
5.1.1 Stress Dependence In Figure 5.1 the stress dependence of the steady-state recoverable compliances and of the steady-state viscosities , calculated from the creep compliances according to Equation 3.3, is quantified for a commercial polypropylene that possesses a linear molecular structure. For an easier comparison, the steady-state elastic compliance is normalized by its linear value and the viscosity by the zero-shear viscosity . Recoverable compliance and viscosity exhibit a linear stress-independent range that is distinctly wider for the viscosity than for the elastic compliance. Furthermore, the elastic compliance shows a much stronger decay with shear stress than the viscosity.
38
5 Dependence of Elastic Quantities on Experimental Parameters
1.4
180°C
1.2 Je/J0e 1
1.4 1.2 1
0.8
0.8
0.6
0.6
0.4
0.4
norm Je, norm
0.2 1
0.2 10
100
1000
[Pa]
10000
Figure 5.1 Normalized stationary viscosities and normalized stationary elastic as functions of constant creep stresses for a commercial polypropylene compliances at 180 °C. is the linear steady-state recoverable compliance and the zero-shear viscosity. Adapted from [5.18]
5.1.2 Temperature Dependence Another important experimental parameter is the temperature. Figure 5.2 presents the creep compliances and the recoverable compliances at = 10 Pa and temperatures of 180, 200, and 220 °C for a commercial linear polypropylene. The experiments have been conducted in the linear range of deformation. The compliance, which is governed by the viscosity at longer times, exhibits a distinct dependincreases with growing temperaence on the temperature in the sense that ture, corresponding to a decreasing viscosity. However, the steady-state recoverable compliance is found to be independent of the temperature. The time-dependent regime of the recoverable compliance reflects a somewhat faster recovery at higher temperatures, as expected from the shorter retardation times due to the lower viscosities.
5.1 Recoverable Compliance
10-1
J(t), J r(tr) [Pa -1]
Polypropylene =10 Pa
10-2
J(t)
10-3
Jr(tr) 10-4
10-5 10-1
180 °C 200 °C 220 °C 100
101
102
103
t, tr [s]
104
Figure 5.2 Creep compliance as a function of creep time and recoverable compliance as a function of recovery time for a commercial polypropylene in the linear range at the temperatures 180, 200, and 220 °C. Reprinted with permission from [5.4]; copyright 2009 American Chemical Society
It is interesting to note that for the curves of and , measured at the different temperatures, master curves can be obtained by shifting them along the time (see Figure 5.3). This means that the examined axis with the same shift factor polypropylene shows a thermorheologically simple behavior. Further details on thermorheological properties of polymer melts can be found in [5.1], for example.
J(t/a T ), J r (t r /a T ) [Pa -1 ]
10-1
Polypropylene =10 Pa T0 =180 °C
10-2
10-3
10-4
180 °C 200 °C 220 °C 10-5 10-1
100
101
102
103
104
t/aT , t r /a T [s]
Figure 5.3 Master curves of and from Figure 5.2 at the reference temperature = 180 °C. Reprinted with permission from [5.4]; copyright 2009 American Chemical Society
39
40
5 Dependence of Elastic Quantities on Experimental Parameters
The independence of the steady-state recoverable compliance from temperature is valid in the nonlinear regime too, as can be concluded from the measurements at a creep stress of 1000 Pa presented in Figure 5.4 for a commercial polypropylene. 10-3 = 1000 Pa
T [°C]
-1
Jr [Pa ]
180 200 220
10-4
PP 10 10
-5
10-1
100
101
102
103
t r [s]
104
Figure 5.4 Recoverable compliance as a function of the recovery time for a commercial polypropylene at different temperatures and a creep stress of 1000 Pa in the nonlinear regime. Reprinted with permission from [5.4]; copyright 2009 American Chemical Society
A steady-state recoverable compliance independent of temperature is reported in [5.2] for a linear low density polyethylene, too, and in [5.3] for polystyrene and polystyrene blends, for example. The existence of master curves for the time dependence of the recoverable compliance is not a general feature of polymer melts, as it becomes obvious from Figis presented at various temperatures for a long-chain branched ure 5.5, where low density polyethylene (LDPE) [5.4]. For a clearer distinction the compliance is plotted on a linear scale, while for the time the logarithmic scale is kept. Evidently, the steady-state values of the recoverable compliances decrease with growing temperatures and consequently a master curve cannot be obtained by a mere shift along the time axis. Such a behavior is found for some other polymer melts too. The temperature dependencies of various polymer melts are discussed in Section 6.4.1.3 in relation to their molecular structure.
5.2 Relaxation Modulus
1.6 1.4
Jr
T 130°C 150°C 170°C 190°C
1.0
-3
-1
Jr(tr) [10 Pa ]
1.2
0.8 0.6 0.4 0.2 0.0 -1 10
LDPE-tub 4 10
0
10
1
2
10 t r [s]
10
3
10
4
10
5
Figure 5.5 Recoverable compliance as a function of the recovery time in a semi-logarithmic plot for a long-chain branched LDPE at four temperatures in the linear regime. Reprinted with permission from [5.4]; copyright 2009 American Chemical Society
5.2 Relaxation Modulus The relaxation modulus of polymer melts is distinctly time-dependent. From the time dependence in the linear range the relaxation spectrum can be determined according to Equation 3.18 or, in the discrete form, from Equation 3.16. Measurements of the relaxation modulus are subject to restrictions with respect to the time window and, subsequently, to the determination of relaxation times at their lower and higher ends. The shortest measurable times are given by the duration of the generation of the deformation step, and the longest times depend on the resolution and noise level of the torque or force transducer. Obviously, the limits are specific of the technical performance of the rheometer used. As an example, in [5.5] a time window between about 0.1 and 30 s is reported for a standard polystyrene characterized with a usual laboratory rotational rheometer. In practice, the relaxation spectra are more often determined from dynamic-mechanical experiments as described in Section 3.3, because the underlying experimental technique and the numerical procedures have been developed to a high standard. Relaxation experiments in the nonlinear regime have gained a special importance, however, to determine the so-called damping function, which makes possible a separation of the time and deformation dependence of the modulus. A review on
41
5 Dependence of Elastic Quantities on Experimental Parameters
damping functions and their applications in constitutive equations is given in [5.6]. The basic concept of time-deformation separability becomes obvious from Figure 5.6. The modulus curves covering about three decades as functions of time decrease with growing deformation steps . But they exhibit very similar shapes as functions of the duration of the relaxation experiments and can be superposed by shifting them along the axis of the modulus on the logarithmic scale. The shift .Taking factors for the different shear steps constitute the damping function , then the relation the curve at the smallest step as the linear reference (5.1) is valid. This experimental finding means that the relaxation modulus dependent divided by the damping function on time and shear can be obtained from describing the nonlinearity.
G(t, ) G(t, )/h( )
42
PS-r-95 T = 200 °C
10 6 Pa 10 5 10 4
G(t, ) / h( )
10 3
x 10
G(t, ) 10 2 10 1
deformation (from top to bottom): 0.1 1.0/1.5/2.0/2.5/3.0/3.5/4.0 4.5 0.1
0.5
1
2 t [s]
Figure 5.6 Relaxation experiments on a standard polystyrene at a temperature of 200 °C and various deformation steps between 0.1 (dotted line) and 4.5 (broken line). represents a master curve obtained by shifting the various modulus curves along the modulus describes the shift factors. For the matter of axis to the curve at a step of = 0.1 [5.7]. convenience and to point out the functionality, the strain steps are designated by instead of and for a clearer presentation is multiplied by 10
For the polystyrene of Figure 5.6 the damping function is plotted in Figure 5.7. At = 1 can be assumed approxsmall steps up to about 0.2 a linear behavior with imately. For higher deformation steps the damping function goes down, but within the accuracy of the measurements it is the same at the two different temperatures 180 and 200 °C. The damping function is not a universal relation and depends on
5.2 Relaxation Modulus
the type of polymer and its molecular structure, as can be seen from [5.7] for polystyrenes and from [5.8] for polyolefins.
h( )
Eq. (5.2) with = 7.8
1
0.1
PS-r-95 (T = 180 °C) PS-r-95 (T = 200 °C) 0
1
2
3
4
5
Figure 5.7 Damping function versus the deformation step for the polystyrene of F igure 5.6 at two temperatures [5.7]. The full line is drawn according to Equation 5.2 with the parameter = 7.8
A general numerical description of the experimental curves based on a molecular theory has not been achieved. Rather, various empirical formulas can be found in the literature, which are reviewed in [5.6]. The full line in Figure 5.7, for example, represents the so-called PSM function (5.2) named after the three persons who proposed it [5.9]. is a fit parameter and was found to be 7.8 for the measurements in Figure 5.7 [5.7]. For some polymers the damping function can be described numerically by the sum of two exponential functions, as shown in [5.10]: (5.3) , , and are material-specific parameters obtained from a numerical description of the experimental damping function with Equation 5.3. The advantage of this exponential damping function is that it can be handled mathematically in an easy way.
43
44
5 Dependence of Elastic Quantities on Experimental Parameters
5.3 Storage Modulus The storage modulus measured in the linear viscoelastic regime may change over several decades in the frequency range applicable for commercial rheometers. An example is given for six anionic polystyrenes with very narrow molar mass distribetween 1.0 and 1.1 in butions characterized by polydispersity indices describes the number average molar mass, and the weight average Figure 5.8. of the samples range between 125 and 750 kg/mol. molar masses
Figure 5.8 Storage modulus as a function of the angular frequency at 180 °C for six anionic polystyrenes with very narrow molar mass distributions and weight average molar masses ranging between 125 and 750 kg/mol from bottom to top
The importance of a wide frequency range becomes obvious from Figure 5.8. In the double-logarithmic plot at low frequencies the terminal regime with the slope of 2 is very pronounced and the linear steady-state quantities and can be determined when making use of the corresponding plot of the loss modulus and the Equations 3.30 and 3.32. At higher frequencies, is only very weakly dependent on frequency, before it starts to rise again. This range is called the rubber-elastic plateau, because a frequency-independent storage modulus is typical of crosslinked rubber (see, e. g., [5.1]). For thermoplastic polymers the rubber-elastic behavior is due to entanglement networks of molecules (see Chapter 7). According to the theory of rubber elasticity, for the plateau modulus, designated as , the relation (5.4)
5.4 Normal Stress Difference
is valid (see [5.19], for example). is the number of cross-links or entanglements per volume, the Boltzmann constant, the absolute temperature, the polymer the molar mass of a molecule strand between two cross-links or entandensity, glements, and the universal gas constant. This relation is the basis for the experimentally determined entanglement molar masses published in [5.20], for example. In [5.1] the relation (5.5) is presented. The factor 2 goes back to the assumption of a fourfold functionality of entanglements. It has to be mentioned, however, that curves of like in Figure 5.8 can be found for narrowly distributed polymers only. For materials with wider molar mass distributions the terminal regime often cannot be reached in the low frequency range attainable and the plateau is often smeared out in such a way that it is difficult to be determined. As it is well known and can be found in many text books on polymer rheology, the temperature dependence of properties of a great number of polymer melts is thermorheologically simple. That means that the modulus curves at different temperatures can be shifted along the logarithmic frequency axis in good approximation to a master curve, while the shift along the modulus axis is negligible. On the other at hand, the existence of such a master curve makes it possible to obtain various temperatures by measuring the frequency dependence at one temperature and shifting this curve with the shift factors known.
5.4 Normal Stress Difference As shown in Figure 4.6, the normal stress difference is a distinct function of time and shear rate. The measurements give a hint that a time-independent value may not be easy to attain. One reason for these difficulties is a fractioning of the polymer melt at the rim of the rheometer plate, particularly at higher shear. With special arrays and experimental care, measurements of steady-state normal stress differences have become possible and are reported in the literature. An example is given in Figure 5.9 for LDPE presenting the normal stress coefficient together with the viscosity [5.10]. The experiments were performed over a wide range of shear rates at the various temperatures indicated in the figure. A master curve for the viscosity function was obtained in the usual way by multiplying
45
46
5 Dependence of Elastic Quantities on Experimental Parameters
the shear rate with the shift factor aT and dividing the viscosity by it. Due to the scaling of n1 with the reciprocal shear rate squared according to Equation 3.39, . The experimentally obthis quantity has consequently been reduced with tained master curve does exhibit a small scatter only and thus represents the normal stress coefficient n1 over a wide range of shear rates or shear stresses, respectively. A constant n1 marking the linear behavior is only found in a narrow range of shear rate. From this plateau, n1 decreases very strongly over five decades as a function of shear rates ranging from 10−3 to 102 s−1. This decay is distinctly more pronounced than that of the viscosity. Similar to the stress dependencies of the viscosity and the recoverable compliance in Figure 5.1, the results of Figure 5.9 confirm a stronger nonlinearity of the elastic in comparison with the viscous behavior. The results of Figure 5.9 show that normal stress coefficients at various temperatures can be derived from the master curve of a thermorheologically simple polymer melt. However, for the stress or rate dependence of n1 a description by a model is not available yet.
Figure 5.9 Steady state values of the viscosity h and the normal stress coefficient n1 obtained from stressing experiments for LDPE. The measurements were performed at the different temperatures indicated and shifted to master curves by using the shift factor aT. The reference temperature was T0 = 150 °C. Adapted from [5.10]
An interesting empirical relation between and reading moduli
and the frequency-dependent
(5.6) is reported in [5.11]. For a typical LDPE it is shown in Figure 5.10 that the validity of this relation is fulfilled with a surprisingly high accuracy, when the shear rate of
5.4 Normal Stress Difference
the stressing experiment is set equal to the angular frequency of the dynamic- mechanical experiment. Equation 5.6 is interesting insofar as it correlates measurements of normal stresses in the nonlinear regime with those of the dynamic moduli in the linear range. Thus, it can be considered as an analog to the Cox-Merz relation that describes the equality of the shear viscosity as a function of the shear rate and the dependence of the magnitude of the complex viscosity on the angular frequency, when shear rate and frequency are chosen the same (see [5.1], for example).
Figure 5.10 Comparison of the first normal stress coefficient as a function of the shear rate with the quantity as a function of the angular frequency , derived from dynamic- mechanical measurements according to Equation 5.6 for LDPE. For an easier comparison, the measured values of the normal stress coefficients are presented as a continuous curve. Adapted from [5.11]
In [5.11] it has been shown that Equation 5.6 is valid for a polystyrene, a high density polyethylene, and a polypropylene too. Because of the purely empirical nature of Equation 5.6, its validity has to be proven for each material separately. Normal stress measurements are often overinterpreted. For example, sometimes Equation 3.41, exactly valid in the linear regime, is applied to determine the recovin the nonlinear range from the first normal stress difference , erable shear according to (5.7) with s being the shear stress. As can be expected, however, this equation may be considered as a rough approximation only (cf. Table 12.1).
47
48
5 Dependence of Elastic Quantities on Experimental Parameters
5.5 Recoverable Elongation As described in Section 4.2.6, the recovery of an elongated polymer sample is a function of time. This time dependence may superpose in a complex way the influences of external parameters like total strain, stress, or temperature. Therefore, in the following, equilibrium data are discussed. In Figure 5.11 the dependence of the steady-state recoverable strain on the total strain is presented for LDPE at three elongational rates and a temperature of 150 °C. It is evident that the recoverable strain significantly depends on the total strain and the strain rate.
Figure 5.11 Steady-state recoverable strain as a function of the total strain for the LDPE IUPAC A at T = 150 °C and various elongational rates. Each symbol marks a separate sample that was stretched up to the different elongations and then recovered by cutting it into pieces according to the method sketched in Section 4.2.6. Adapted from [5.17]
For all three elongational rates chosen, increases with and reaches a plateau value that becomes higher the larger the elongational rate . The broken line in Figure 5.11 marks the case for a total reversibility of the elongation applied. At high elongational rates and rather small strains or short times, respectively, most of the deformation is reversible, but at higher elongations the deformation contains a significant viscous portion. In Figure 5.12 the recoverable strains for polystyrene as a function of the tensile stress at 130 °C are presented. describes the recoverable strain in the regime where it is independent of the total deformation. The recoverable strain increases
5.5 Recoverable Elongation
with the stress applied and then starts to level off. This behavior indicates that the recoverable part of the elongation approaches a limiting value that can be assumed to be reached when all molecules are stretched. The steady-state recoverable compliance defined as (5.8) is presented in Figure 5.12 too. This material function is constant at small stresses, designating the linear range of deformation, and decreases at higher stresses, marking the nonlinear regime.
rs
10-4
Polystyrene T=130°C
De
De [Pa-1]
0
10
10-5
10-1
rs
10-6
10-2
10-7
E=const. . =const. 10-3
103
104
105
106
E [Pa]
107
10-8
Figure 5.12 Steady-state recoverable strain and recoverable tensile compliance as functions of the tensile stress for polystyrene at 130 °C. The triangles mark results from experiments at constant stress, the circles those at constant strain rate. Adapted from [5.15]
It is worth noting that the steady-state values of the recoverable strain from experiments at constant stresses (creep experiments) and those at constant elongational rates (stressing experiments) come to lie on the same curve, when elongational rate is converted to tensile stress by making use of the elongational viscosity measured. This result is very reasonable, because the conformation of the molecules in an equilibrium state should not be dependent on the way in which it is attained. The agreement stands for the reliability of the two experimental modes. Another external parameter is the temperature. In Figure 5.13 the steady-state refor LDPE is shown in a temperature range between 125 and coverable strain 180 °C. The samples were elongated at various constant stresses [5.17]. Within the experimental accuracy the recoverable strain is independent of the temperatures chosen. This behavior is different from the significant temperature dependence of
49
50
5 Dependence of Elastic Quantities on Experimental Parameters
the elongational viscosity, as shown in [5.17], for example. As seen from Figure 5.13, the recoverable strain of the LDPE increases with tensile stress, similar to the behavior of the PS presented in Figure 5.12.
Figure 5.13 Steady-state recoverable strain at various temperatures for LDPE. The previous elongations were performed at the indicated constant tensile stresses up to a stretching ratio of 40, corresponding to a Hencky strain of 3.7 at which a steady state of creep was achieved in all cases. Adapted from [5.17]
5.6 Extrudate Swell 5.6.1 General Features of Extrudate Swell The extrudate swell is a complex function of various external parameters. This fact is obvious from Figure 5.14, which comprehensively presents the various dependaccording to encies [5.1]. In this figure the extrudate swell, defined as of the capEquation 3.43, is plotted as a function of the length to radius ratio illary, for samples of a commercial polystyrene annealed in a silicone oil bath after extrusion. The general feature is that the extrudate swell decreases with the capillary length from comparatively high values obtained for very short capillaries and , which is smaller by a factor of about 2 seems to approach a plateau for large in comparison with an orifice. Similar results for the extrudate swell have been reported for other polymer melts too. In [5.12], for example, the extrudate swell of
5.6 Extrudate Swell
a high density polyethylene has been presented at different shear rates for capillarto . The extrudate swell increases with growies ranging from and approaches a plateau at larger capillary ing shear rate. It decreases with lengths. The existence of a significant extrudate swell for an orifice of very small length that can be deduced from Figure 5.14 indicates the role of the entrance flow for this quantity. As measured by laser Doppler velocimetry, large elongational components may occur in the entrance region of a capillary [5.13]. In the case of visco elastic materials, some portion of the deformation is elastic and recovers when the tensile stress becomes zero within the capillary. This recoil of the elongated molecules should be more pronounced the longer their way through the capillary, that is, the larger the capillary length at a given throughput.
Figure 5.14 Extrudate swell of a commercial polystyrene as defined by Equation of capillaries at two different stresses 3.43, as a function of the length to radius ratio at the wall and three extrusion temperatures . After extrusion, the samples were annealed in a silicone oil bath to reach their equilibrium. Adapted from [5.1]
It is a well-known feature, however, that there is still a remarkable extrudate swell measurable even for long capillaries, when the elastic deformation generated by the entrance flow should have recovered within the die. This effect goes back to the elastic portion of the shear in the capillary due to normal stresses. The release of the exiting strand from the geometrical constraints within the capillary leads to the recovery of the reversible deformation during shear and of the orientations left from the stretching of molecules in the entrance region of the capillary. The recovery of these two parts outside the capillary results in the observed extrudate swell.
51
52
5 Dependence of Elastic Quantities on Experimental Parameters
From Figure 5.14 it is evident that within the accuracy of the measurements the . This means that, for the polyextrudate swell does not change much for styrene studied, the elastic deformation exerted by the elongational deformation in the entrance region of the die used has been recovered to a great deal by passing the die, and the extrudate swell is mainly determined, then, by normal stress effects. At a given die geometry this recovery process depends, of course, on the retardation times of the melt and thus on molar mass and temperature. From Figure 5.14 it follows that the extrudate swell increases with the shear stress at the wall, determined according to Equation 4.4, and that it is independent of temperature within the range chosen. However, due to the complexity of extrudate swell, these apparently obvious results have to be discussed in some more detail.
5.6.2 Detailed Analysis of Extrudate Swell Rather easy to analyze qualitatively is the stress dependence of the extrudate swell for short capillary lengths where the elongational deformation exerted at the die entrance is dominant. A result of the deformation of a melt in the entrance region is the so-called entrance pressure loss. As shown in [5.1] for polystyrene and visualized in Figure 12.2, this quantity increases with the shear rate and, consequently, the shear stress at the wall. For linear polymers, the entrance pressure loss is primarily due to extensional stresses [5.14]. Therefore, higher extensional stresses in the entrance region of a capillary are related to higher shear stresses at the wall. The higher stresses go along with larger extensions in the entrance region that lead to an increase in the recoverable elongation , as demonstrated in Figure 5.12 and Figure 5.13, for example, and consequently to a larger extrudate swell in the case of short capillaries, for which recovery processes of the entrance flow within the die are negligible. The independence from temperature of the extrudate swell in Figure 5.14 corresponds to that of the recoverable strain of the at small LDPE in Figure 5.13 and that reported in [5.15] for two polystyrenes. For long capillaries, in which the elastic deformation exerted in the entrance region may totally recover, recoverable shear becomes the source of elasticity. Also in this case the results of the extrudate swell presented in Figure 5.14 remain in qualitative accordance with the known dependencies of elastic quantities on external parameters, because the recoverable shear increases with stress as can be deduced from Figure 5.1 and is independent of temperature (see Figure 5.4), similar to the corresponding quantities in elongational flow according to Figure 5.12 and Figure 5.13. More difficult to analyze is the extrudate swell for capillaries where the recovery of the elastic part of elongation and the elastic part of shear superimpose. Comparable measurements of the extrudate swell of different polymers are frequently per-
5.6 Extrudate Swell
formed with a given capillary at constant apparent shear rates. Characterizations of this kind are not very meaningful for the following reason: The constant apparent shear rates result in the same throughput times for a chosen capillary, of course, and, consequently, comparable time windows for the viscoelastic effects. But different viscosities lead to different stresses. The consequence is that elastic properties changing with stress and their effect on the extrudate swell have to be taken into account. Additionally, the sample with the higher viscosity relaxes more slowly due to the inherent longer relaxation time. Choosing a constant stress at the capillary wall is the other experimental mode that can be performed. Then, for a given capillary, the apparent shear rates and throughput times change according to the viscosity of the sample. Consequently, the recovery within the capillary may be different and, following from that, the extrudate swell at the exit too. Assuming that for two samples the same elastic deformation is imposed at the capillary entrance, the material with the lower viscosity has the larger apparent shear rate and the shorter throughput time at comparable stresses. Therefore, even if the relaxation times of the samples were the same, the recovery within the capillary would be slower for the sample of low viscosity. But because of the relaxation time of the material becoming shorter with decreasing viscosity, the recovery is faster at comparable throughput times and the two effects may compensate each other to some extent. This discussion demonstrates the complexity regarding the detailed interpretation of extrudate swell measurements with respect to rheological properties.
5.6.3 Extrudate Swell for Various Die Geometries The discussion above makes it obvious that the shape of a die may have a significant effect on the extrudate swell. These dependencies are interesting for the assessment of the role of extrudate swell in cases of processing, as discussed in Sections 12.4 and 12.5. Some tendencies can be derived from results of laboratory experiments on LDPE reported in [5.16]. At a constant throughput, the extrudate swell of a tapered capillary with an entrance radius of 6 mm and an exit radius of 1 mm was compared with that of the same tapered die, but with a circular capillary of a radius of 1 mm and a length of 20 mm mounted at the exit. The extrudate swell measured by the ratio of the radius of the extruded strand to that of the capillary at its exit decreased by about 20 %. This finding is to be expected, due to the partial recoil of the molecules in the circular capillary that were previously stretched during the extensional flow in the tapered section. The extrudate swell could be further reduced by a trumpet-shaped outlet with a radius of 2 mm at the exit. This geometry enables a partial recoil of the molecules before the free total recovery after exiting the die resulting in the decreased extrudate swell. This example shows
53
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5 Dependence of Elastic Quantities on Experimental Parameters
how complex the assessment of the extrudate swell in a real extrusion process may become, because a melt often passes various geometries in a tool. Furthermore, the shear stresses within a real tool may change due to varying geometries. Moreover, another feature comes into play for an assessment of the effect of extrudate swell in processing. In real extrusion processes the equilibrium data presented in Figure 4.7 are seldom reached, because for practical reasons the extruded item is quickly cooled and subsequently the molecular motions are frozen when the melt exits the die. If a calibration step is used within an extrusion process, free expansion of a strand is not possible, and in the case of an applied pulling force a tension counteracts the extrudate swell (see Section 12.7). Two other factors have to be considered additionally and aggravate a quantitative description of extrudate swell even in a simple circular die. The shear stress or shear rate, respectively, are not constant across the cross section of a capillary and the elongational flow in the entrance region may be of a complex multiaxial nature. Regarding the stresses in Figure 5.14 at which capillary rheometry is usually undertaken, these are much higher than in the linear range of the creep recovery experiments presented in Figure 4.3, for example. Thus, it has to be assumed that all the extrudate swell experiments are performed in the nonlinear range of deformation. This feature makes a quantitative correlation with molecular data difficult, because melt properties change with stress and, thus, characteristic quantities are not available.
5.7 References [5.1]
Münstedt H., Schwarzl F. R., Deformation and Flow of Polymeric Materials, Springer, Berlin (2014)
[5.2]
Gabriel C., Münstedt H., Creep recovery behavior of metallocene linear low density polyethylenes, Rheol. Acta 38 (1999), 393–403
[5.3]
Orbon S. J., Plazek D. J., Recoverable compliance of a series of bimodal molecular blends of polystyrene, J. Polym. Sci. B 17 (1979), 1871–1890
[5.4]
Resch J. A., Stadler F. A., Kaschta J., Münstedt H., Temperature dependence of the linear steadystate compliance of linear and long-chain branched polyethylenes, Macromolecules 42 (2009), 5676–5683
[5.5]
Hepperle J., Münstedt H., Rheological properties of branched polystyrenes: nonlinear shear and extensional behavior, Rheol. Acta 45 (2006), 717–727
[5.6]
Rolon-Garrido V. H., Wagner M. H., The damping function in rheology, Rheol. Acta 48 (2009), 245–284
[5.7]
Hepperle J., Einfluss der molekularen Struktur auf rheologische Eigenschaften von Polystyrol- und Polycarbonatschmelzen (Influence of the molecular structure on rheological properties of poly styrene and polycarbonate melts), Doctoral Thesis, University Erlangen-Nuremberg, Shaker Verlag, Aachen (2002)
5.7 References
[5.8]
Stadler F. J., Auhl D., Münstedt H., Influence of the molecular structure of polyolefins on the damping function in shear, Macromolecules 41 (2008), 3720–3726
[5.9]
Papanastasiou A. C., Scriven L. E., Macosko C. W., An integral constitutive equation for mixed flows: Viscoelastic characterization, J. Rheol. 27 (1983), 387–410
[5.10] Laun H. M., Description of the non-linear shear behaviour of a low density polyethylene melt by means of an experimentally determined strain dependent memory function, Rheol. Acta 17 (1978), 1–15 [5.11]
Laun H. M., Prediction of elastic strains of polymer melts in shear and elongation, J. Rheol. 30 (1986), 459–501
[5.12] Han C. D., Charles M., Philippoff W., Rheological implications of exit pressure and die swell in steady capillary flow of polymer melts. I. Primary normal stress difference and the effect of L/D ratio on elastic properties, Trans. Soc. Rheol. 14 (1970), 393–408 [5.13] Wassner E., Schmidt M., Münstedt H., Entry flow of a low-density polyethylene melt into a slit die: An experimental study by laser-Doppler velocimetry, J. Rheol. 43 (1999), 1339–1353 [5.14] Münstedt H., Schwetz M., Heindl M., Schmidt M., Influence of molecular structure on secondary flow of polyolefins as investigated by laser-Doppler velocimetry, Rheol. Acta 40 (2001), 384–394 [5.15] Münstedt H., New universal extensional rheometer for polymer melts. Measurements on a polystyrene sample, J. Rheol. 23 (1979), 421–436 [5.16] Laun H. M., Schuch H., Transient elongational viscosities and drawability of polymer melts, J. Rheol. 33 (1989), 119–175 [5.17]
Münstedt H., Laun H. M., Elongational behaviour of a low density polyethylene melt. II. Transient behavior in constant stretching rate and tensile creep experiments. Comparison with shear data. Temperature dependence of the elongational properties, Rheol. Acta 18 (1979), 492–504
[5.18] Resch J. A., Kaschta J., Wolff F., Münstedt H., Influence of molecular parameters on the stress dependence of viscous and elastic properties of polypropylenes in shear, Rheol. Acta 50 (2011), 53–63 [5.19] Ferry J. D., Viscoelastic Properties of Polymers, John Wiley, New York (1980) [5.20] Fetters L. J., Lohse D. J., Colby R. H. in: Mark J. E. (ed.), Physical Properties of Polymers Handbook, Springer, Berlin (2005)
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