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Principles of the Determination of Elastic Properties
3
Principles of the Determination of Elastic Properties
3.1 Creep Recovery Experiment and Retardation Spectrum Elastic properties manifest themselves in many effects. In principle, each of them could be used to get insight into the elastic behavior of a material. A rather obvious feature of elastic properties is the recovery of some of the deformation previously exerted on a sample, when the deforming stress is released. For quantitative measurements, the deformation and the recovery as well have to be performed under certain conditions. A convenient method is creep and the subsequent creep recovery. The principle of such a test is sketched in Figure 3.1. At the time t = 0 a sample is loaded by a constant stress and the deformation registered as a function of time. After the creep time t0 the stress is set to zero. For a viscoelastic sample, some of the total deformation recovers before attaining a time-independent level. This plateau is due to the irreversible viscous deformation of the sample and the recoverable portion has its origin in the elastic behavior of the material.
Figure 3.1 Schematic representation of a creep and creep recovery experiment
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3 Principles of the Determination of Elastic Properties
The deformation itself depends on material properties and on external parameters. Therefore, in the case of creep experiments, the compliance defined as the deformation related to the stress is discussed as the corresponding material-specific function. In Figure 3.1 the creep compliance (3.1) is plotted, with the variable being the constant shear stress applied and the resulting total shear. According to the Boltzmann superposition principle the creep compliance can be written as (see, e. g., [3.1]) (3.2) is the instantaneous compliance, the time-dependent recoverable complithe viscosity. This additivity is visualized in Figance or creep function, and ure 3.1. It should be mentioned, however, that the superposition principle was derived by Boltzmann for the linear case, in which the creep function and the viscosity are independent of the stress applied. Many experimental results have shown that the decomposition of the creep compliance as given by Equation 3.2 can be used in the nonlinear regime too. Because the instantaneous compliance is comparatively small and the creep function attains a constant value after some time, for long enough creep times the viscous part dominates the compliance and the viscosity can be obtained from Equation 3.2: (3.3) For correspondingly small stresses in the linear regime, the zero-shear viscosity follows as (3.4) In the linear regime the creep function can be described numerically by (3.5) with being discrete retardation times and the corresponding retardation strengths. For a continuous retardation spectrum it follows:
3.1 Creep Recovery Experiment and Retardation Spectrum
(3.6) with the retardation function
.
The recoverable portion of shear and therewith the recoverable compliance as functions of the recovery time are dependent in general on the stress and the time of the preceding creep; that means (3.7) Because the recoverable compliance is defined as the difference between the total compliance and the viscous portion, Equation 3.7 can be written as (3.8) For polymer melts and solutions the time-dependent recoverable compliance attains a constant value at sufficiently long creep and recovery times, and one gets for the steady-state recoverable compliance: (3.9) and for the linear steady-state recoverable compliance: (3.10) This quantity is of great interest insofar as it is related to the retardation spectrum by (3.11) where designates the retardation time. For a discrete retardation spectrum one can write, according to Equation 3.5, (3.12) with being the retardation strengths. More detailed information on the derivation of these relations can be found in [3.1], for example. For polymer melts the
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3 Principles of the Determination of Elastic Properties
instantaneous compliance is negligibly small in comparison with the time- dependent recoverable compliance. Thus, from creep recovery experiments in the linear range the retardation spectrum can be determined directly, which is a fingerprint of molecular motions. That is the reason why measurements of the recoverable compliance have found much attention recently as a rheological tool for the characterization of polymers. Examples are given in Section 10.3.3. In the linear steady-state regime the creep compliance can be written as (3.13) where (3.14) is defined as the longest retardation time according to the Voigt-Kelvin model (see, e. g., [3.1]), with being the zero-shear viscosity. Three distinct times are used to discuss creep recovery: describes the time scale of the experiment and is assigned to the duration of the creep experiment. Then, as it is obvious from Figure 3.1 and Figure 3.2. the recovery time follows As a matter of convention, for recovery and creep the same time scales are used, and then creep compliance and recoverable compliance can be compared conveniently.
Figure 3.2 Different representations of a creep experiment
As sketched in Figure 3.2, the strain increases during creep and consequently the compliance as the corresponding material-specific function shows a similar behavior. During recovery the total strain decreases, but the recoverable strain becomes larger. Thus, the recoverable compliance is a material-specific function increasing with time and, therefore, it is reasonable to present on the same scale
3.2 Relaxation Experiment and Relaxation Spectrum
as . The result of these considerations is presented in the bottom left diagram as linear plots. For times and compliances covering wide ranges a double-logarithmic plot is often chosen. Logically, the appearances of the curves change in the way as schematically shown in the bottom right diagram. The linear or double-logarithmic and presented in the bottom diagrams of Figure 3.2 are applied plots of throughout this book.
3.2 Relaxation Experiment and Relaxation Spectrum Another experiment that reflects the viscoelasticity of a material is the measurement of its relaxation behavior. It can be seen as the counterpart to creep, because a deformation is imposed on a sample and the time dependence of the resulting stress registered. The principle of a relaxation experiment is sketched in Figthe constant deformation is applied and then kept over ure 3.3. At the time the time of the experiment. The corresponding stress rises instantaneously and gradually decreases subsequently. In the case of a viscoelastic liquid the stress approaches zero; for a viscoelastic solid a finite value is attained after longer times.
Figure 3.3 Schematics of a relaxation experiment. is the instantaneous modulus and stands for the time-dependent modulus in the linear case
11
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3 Principles of the Determination of Elastic Properties
The stress is dependent on the deformation step and material properties. Therefore, the modulus (3.15) is considered as the material-specific function resulting from the relaxation experiment. In the linear range of deformation the modulus is independent of the instantaneously applied shear and can be written as (3.16) with (3.17) For viscoelastic liquids, becomes zero. Therefore, this quantity is not discussed further in this presentation on polymer melts, which can be considered as visco elastic liquids. The numerical description given by Equation 3.16 is equivalent to the Maxwell model (see Section 7.1). The describe the discrete relaxation times and the the relaxation strengths. for various represent the discrete relaxation spectrum. For a continuous spectrum of relaxation times, the sum in Equation 3.16 changes to an integral and one gets (3.18) with
representing the continuous relaxation spectrum.
3.3 Dynamic-Mechanical Experiment A widely used method to determine elastic properties in the linear range of deformation are dynamic-mechanical experiments. They are described in many textbooks on rheology and, therefore, are discussed only shortly here. A viscoelastic sample may be exposed to a sinusoidal shear deformation (3.19)
3.3 Dynamic-Mechanical Experiment
Then, in the case of a linear behavior the stress follows as (3.20) w is the angular frequency, d the phase shift between strain and stress. and are the amplitudes of shear strain and stress, respectively. Defining a modulus according to (3.21) Equation 3.20 reads (3.22) This equation can be rewritten introducing (3.23) as the modulus in phase with the exciting strain and (3.24) being the modulus shifted by 90°. They correspond to a storage modulus and a loss modulus, respectively, and can be interpreted as the real part, G¢, and the imaginary part, G¢¢, of a complex modulus (3.25) with (3.26) From these relations it is easily seen that (3.27) and are related to the relaxation spectrum e. g., [3.1])
of a Maxwell model by (see,
(3.28)
(3.29)
13
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3 Principles of the Determination of Elastic Properties
For , that is, in the so-called terminal regime, one gets the following relations with the well-defined rheological quantities and , by making use of according to Equation 3.14 (see [3.1], for example): (3.30)
(3.31) Of particular interest for this book is the elastic quantity , which obviously does not follow directly from in the terminal regime. Rather, the relation (3.32) has to be considered, when elastic properties from dynamic-mechanical measurements are discussed. A direct access to from dynamic-mechanical experiments follows from the determination of the complex compliance defined by (3.33) For the real part
, one gets (see [3.1]) (3.34)
and for the terminal regime, it follows from Equation 3.11: (3.35) Between the complex modulus and compliance and their real and imaginary parts, relatively simple relations exist that may be helpful for certain conversions. From and it follows . By comparison of the real and imaginary parts of the complex quantities simple relations between compliance and modulus can be obtained. For example, the real part of the complex compliance reflecting elastic properties is related to the complex modulus by (3.36)
3.4 Stressing Experiment
and, vice versa, the relation (3.37) holds.
3.4 Stressing Experiment As described in Section 2.1, one feature of elasticity in polymer liquids is the occurrence of normal forces, which manifests itself by the climbing of a material up a rotating rod, for example. This effect may be quantified by measuring the normal force that a viscoelastic material between two coaxial plates with the area exerts when they are revolved with respect to each other at the constant shear rate . The first normal stress difference is given, then, by (3.38) and being diagonal components of the stress tensor (see, e. g., [3.2]). As with the relevant material-specific quantity, the normal stress coefficient is considered, which is defined as (3.39) and becomes (3.40) in the linear steady-state regime (see [3.1]). The factor 2 follows from Lodge’s theory of elastic liquids [3.3]. As can be seen from Equation 3.40, the normal stress coefficient in the linear range is a function of the linear steady-state recoverable compliance and the zero-shear viscosity . The elastic compliance , which is of primary interest when elastic properties are discussed, can be obtained by (3.41) with
.
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3 Principles of the Determination of Elastic Properties
In principle, from the normal stress and the shear stress measurements performed in parallel in the linear and steady-state regime of a stressing experiment, the elastic compliance as a characteristic quantity of the elastic properties of a melt can be determined. From Equation 3.40 it becomes obvious that normal stress differences alone may be used as a comparative qualitative measure of elasticity only for samples that have comparable viscosities, even in the range of linear viscoelasticity.
3.5 Capillary Rheometry The elastic properties of a polymer melt become obvious from the swell of a strand extruded through a die, as sketched in Figure 3.4. The shape of the extrudate depends on the geometry of the die and, therefore, is generally not easy to quantify. For studying the extrudate swell, circular capillaries are applied as shown in Figure 3.4. The measured quantity is the diameter of the exiting strand . The extrudate swell is defined as (3.42) or (3.43) with
being the diameter of the capillary.
The definition of Equation 3.43 is preferentially used in this book, because it directly describes the swell by the relative increase of the diameter and is physically is often found for the extrumore descriptive. In the literature the definition date swell. The extrudate swell is a very obvious manifestation of the elastic properties of a melt, but its interpretation is complex as sketched in Section 2.2. Furthermore, some care has to be taken to get well-defined reliable data, because the strand may be frozen before the equilibrium of the recovery is reached (see Section 4.2.5). Moreover, the linear regime is difficult to attain due to the pressures and shear stresses, respectively, necessary for reliable measurements. This fact and the dependence of the extrudate swell on geometrical features of the die and the flow fields in its entry region make it necessary to be aware of the complexity of the extrudate swell as a measure of elastic properties. Nevertheless, the extrudate swell offers some insight into the role of elasticity for processing, as described in Chapter 12.
3.5 Capillary Rheometry
p
d0 L
d
Figure 3.4 Schematics of the swell of a polymer melt extruded by the pressure through a and the length . is the diameter of the exiting strand die with the radius
An effect in capillaries that is due to elastic properties of polymer melts is the so. It occurs in capillaries, when one of two pressure called hole pressure transducers facing each other is not flush-mounted [3.4]. is the pressure measured by a gauge flush-mounted to the wall of a slit capillary and the pressure at the bottom of a hole in the wall just opposite the position of the other transducer. Detailed calculations of various flow patterns around holes are given in [3.5]. For the hole shaped as a slot perpendicular to the flow direction, from measurements of the hole pressure as a function of the shear stress at the wall, the first normal follows (see [3.6]): stress difference (3.44) In principle, this method makes it possible to measure normal stress differences at shear stresses higher than those in rotational rheometry and thus closer to processing conditions. But the experimental difficulties, particularly with the pressure gauges in the case of polymer melts, have hindered a widespread practical use so far.
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3 Principles of the Determination of Elastic Properties
3.6 Recoverable Elongation A very obvious effect of an elastic deformation is the shape recovery of a uniaxially stretched sample after setting the previously applied tensile stress to zero. In the case of a metal spring the total elongation recovers spontaneously; a cross-linked rubber shows a complete recovery too, but with some time lag; and a polymer melt may only exhibit a time-dependent partial rebound of the extended sample. In the Cauchy measure the strain exerted on a sample in a uniaxial tensile experiment is (3.45) and the recoverable strain follows as (3.46) where is the length of the stretched sample and the initial sample length. is the length of the retarded sample, which is a function of time for viscoelastic materials and reaches an equilibrium value after long enough retardation times. For larger deformations, as in the case of rubber or polymer melts, the Hencky strain (3.47) is commonly used and the corresponding recoverable portion reads (3.48) is the total stretching ratio and
the recoverable stretching ratio.
Two limiting cases may be discussed to elucidate Equation 3.48, which is not so common in rheology. When the length after retardation is the same as that of the elongated sample , then evidently the recoverable elongation is zero. Attaining the length of the initial sample after retardation obviously means that the reversible deformation is equal to the total elongation. The strains in the Cauchy and Hencky measures approach each other when the total or recoverable deformations of a sample are small. For the characterization of material-specific properties the compliances are used. In the case of the Hencky measure used for polymer melts, they are defined as (3.49)
3.7 References
for the compliance and (3.50) for the recoverable compliance in elongation, with being the tensile stress. For the matter of convenience, the index H is omitted for the compliances determined in elongation. The steady-state elastic compliance is defined as (3.51) and becomes in the linear regime of deformation. Using the Cauchy measure for the determination of recoverable tensile compliances leads to values different from those in the Hencky measure. They become comparable for small recoverable strains, when the Hencky strain approaches the Cauchy strain. and
are related to each other by (3.52)
3.7 References [3.1]
Münstedt H., Schwarzl F. R., Deformation and Flow of Polymeric Materials, Springer, Berlin (2014)
[3.2]
Macosko C. W., Rheology: Principles, Measurements, and Applications, Wiley-VCH, New York (1994)
[3.3]
Lodge A. S., Elastic Liquids, Academic Press, New York (1964)
[3.4]
Higashitani K., Lodge A. S., Hole pressure error measurements in pressure-generated shear flow, Trans. Soc. Rheol. 16 (1972), 307–335
[3.5]
Higashitani K., Pritchard W. G., A kinematic calculation of intrinsic errors in pressure measurements made with holes, Trans. Soc. Rheol. 16 (1972), 687–696
[3.6]
Ferry J. D., Viscoelastic Properties of Polymers, John Wiley, New York (1980)
19
Principles of the Determination of Elastic Properties
3.1 Creep Recovery Experiment and Retardation Spectrum Elastic properties manifest themselves in many effects. In principle, each of them could be used to get insight into the elastic behavior of a material. A rather obvious feature of elastic properties is the recovery of some of the deformation previously exerted on a sample, when the deforming stress is released. For quantitative measurements, the deformation and the recovery as well have to be performed under certain conditions. A convenient method is creep and the subsequent creep recovery. The principle of such a test is sketched in Figure 3.1. At the time t = 0 a sample is loaded by a constant stress and the deformation registered as a function of time. After the creep time t0 the stress is set to zero. For a viscoelastic sample, some of the total deformation recovers before attaining a time-independent level. This plateau is due to the irreversible viscous deformation of the sample and the recoverable portion has its origin in the elastic behavior of the material.
Figure 3.1 Schematic representation of a creep and creep recovery experiment
8
3 Principles of the Determination of Elastic Properties
The deformation itself depends on material properties and on external parameters. Therefore, in the case of creep experiments, the compliance defined as the deformation related to the stress is discussed as the corresponding material-specific function. In Figure 3.1 the creep compliance (3.1) is plotted, with the variable being the constant shear stress applied and the resulting total shear. According to the Boltzmann superposition principle the creep compliance can be written as (see, e. g., [3.1]) (3.2) is the instantaneous compliance, the time-dependent recoverable complithe viscosity. This additivity is visualized in Figance or creep function, and ure 3.1. It should be mentioned, however, that the superposition principle was derived by Boltzmann for the linear case, in which the creep function and the viscosity are independent of the stress applied. Many experimental results have shown that the decomposition of the creep compliance as given by Equation 3.2 can be used in the nonlinear regime too. Because the instantaneous compliance is comparatively small and the creep function attains a constant value after some time, for long enough creep times the viscous part dominates the compliance and the viscosity can be obtained from Equation 3.2: (3.3) For correspondingly small stresses in the linear regime, the zero-shear viscosity follows as (3.4) In the linear regime the creep function can be described numerically by (3.5) with being discrete retardation times and the corresponding retardation strengths. For a continuous retardation spectrum it follows:
3.1 Creep Recovery Experiment and Retardation Spectrum
(3.6) with the retardation function
.
The recoverable portion of shear and therewith the recoverable compliance as functions of the recovery time are dependent in general on the stress and the time of the preceding creep; that means (3.7) Because the recoverable compliance is defined as the difference between the total compliance and the viscous portion, Equation 3.7 can be written as (3.8) For polymer melts and solutions the time-dependent recoverable compliance attains a constant value at sufficiently long creep and recovery times, and one gets for the steady-state recoverable compliance: (3.9) and for the linear steady-state recoverable compliance: (3.10) This quantity is of great interest insofar as it is related to the retardation spectrum by (3.11) where designates the retardation time. For a discrete retardation spectrum one can write, according to Equation 3.5, (3.12) with being the retardation strengths. More detailed information on the derivation of these relations can be found in [3.1], for example. For polymer melts the
9
10
3 Principles of the Determination of Elastic Properties
instantaneous compliance is negligibly small in comparison with the time- dependent recoverable compliance. Thus, from creep recovery experiments in the linear range the retardation spectrum can be determined directly, which is a fingerprint of molecular motions. That is the reason why measurements of the recoverable compliance have found much attention recently as a rheological tool for the characterization of polymers. Examples are given in Section 10.3.3. In the linear steady-state regime the creep compliance can be written as (3.13) where (3.14) is defined as the longest retardation time according to the Voigt-Kelvin model (see, e. g., [3.1]), with being the zero-shear viscosity. Three distinct times are used to discuss creep recovery: describes the time scale of the experiment and is assigned to the duration of the creep experiment. Then, as it is obvious from Figure 3.1 and Figure 3.2. the recovery time follows As a matter of convention, for recovery and creep the same time scales are used, and then creep compliance and recoverable compliance can be compared conveniently.
Figure 3.2 Different representations of a creep experiment
As sketched in Figure 3.2, the strain increases during creep and consequently the compliance as the corresponding material-specific function shows a similar behavior. During recovery the total strain decreases, but the recoverable strain becomes larger. Thus, the recoverable compliance is a material-specific function increasing with time and, therefore, it is reasonable to present on the same scale
3.2 Relaxation Experiment and Relaxation Spectrum
as . The result of these considerations is presented in the bottom left diagram as linear plots. For times and compliances covering wide ranges a double-logarithmic plot is often chosen. Logically, the appearances of the curves change in the way as schematically shown in the bottom right diagram. The linear or double-logarithmic and presented in the bottom diagrams of Figure 3.2 are applied plots of throughout this book.
3.2 Relaxation Experiment and Relaxation Spectrum Another experiment that reflects the viscoelasticity of a material is the measurement of its relaxation behavior. It can be seen as the counterpart to creep, because a deformation is imposed on a sample and the time dependence of the resulting stress registered. The principle of a relaxation experiment is sketched in Figthe constant deformation is applied and then kept over ure 3.3. At the time the time of the experiment. The corresponding stress rises instantaneously and gradually decreases subsequently. In the case of a viscoelastic liquid the stress approaches zero; for a viscoelastic solid a finite value is attained after longer times.
Figure 3.3 Schematics of a relaxation experiment. is the instantaneous modulus and stands for the time-dependent modulus in the linear case
11
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3 Principles of the Determination of Elastic Properties
The stress is dependent on the deformation step and material properties. Therefore, the modulus (3.15) is considered as the material-specific function resulting from the relaxation experiment. In the linear range of deformation the modulus is independent of the instantaneously applied shear and can be written as (3.16) with (3.17) For viscoelastic liquids, becomes zero. Therefore, this quantity is not discussed further in this presentation on polymer melts, which can be considered as visco elastic liquids. The numerical description given by Equation 3.16 is equivalent to the Maxwell model (see Section 7.1). The describe the discrete relaxation times and the the relaxation strengths. for various represent the discrete relaxation spectrum. For a continuous spectrum of relaxation times, the sum in Equation 3.16 changes to an integral and one gets (3.18) with
representing the continuous relaxation spectrum.
3.3 Dynamic-Mechanical Experiment A widely used method to determine elastic properties in the linear range of deformation are dynamic-mechanical experiments. They are described in many textbooks on rheology and, therefore, are discussed only shortly here. A viscoelastic sample may be exposed to a sinusoidal shear deformation (3.19)
3.3 Dynamic-Mechanical Experiment
Then, in the case of a linear behavior the stress follows as (3.20) w is the angular frequency, d the phase shift between strain and stress. and are the amplitudes of shear strain and stress, respectively. Defining a modulus according to (3.21) Equation 3.20 reads (3.22) This equation can be rewritten introducing (3.23) as the modulus in phase with the exciting strain and (3.24) being the modulus shifted by 90°. They correspond to a storage modulus and a loss modulus, respectively, and can be interpreted as the real part, G¢, and the imaginary part, G¢¢, of a complex modulus (3.25) with (3.26) From these relations it is easily seen that (3.27) and are related to the relaxation spectrum e. g., [3.1])
of a Maxwell model by (see,
(3.28)
(3.29)
13
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3 Principles of the Determination of Elastic Properties
For , that is, in the so-called terminal regime, one gets the following relations with the well-defined rheological quantities and , by making use of according to Equation 3.14 (see [3.1], for example): (3.30)
(3.31) Of particular interest for this book is the elastic quantity , which obviously does not follow directly from in the terminal regime. Rather, the relation (3.32) has to be considered, when elastic properties from dynamic-mechanical measurements are discussed. A direct access to from dynamic-mechanical experiments follows from the determination of the complex compliance defined by (3.33) For the real part
, one gets (see [3.1]) (3.34)
and for the terminal regime, it follows from Equation 3.11: (3.35) Between the complex modulus and compliance and their real and imaginary parts, relatively simple relations exist that may be helpful for certain conversions. From and it follows . By comparison of the real and imaginary parts of the complex quantities simple relations between compliance and modulus can be obtained. For example, the real part of the complex compliance reflecting elastic properties is related to the complex modulus by (3.36)
3.4 Stressing Experiment
and, vice versa, the relation (3.37) holds.
3.4 Stressing Experiment As described in Section 2.1, one feature of elasticity in polymer liquids is the occurrence of normal forces, which manifests itself by the climbing of a material up a rotating rod, for example. This effect may be quantified by measuring the normal force that a viscoelastic material between two coaxial plates with the area exerts when they are revolved with respect to each other at the constant shear rate . The first normal stress difference is given, then, by (3.38) and being diagonal components of the stress tensor (see, e. g., [3.2]). As with the relevant material-specific quantity, the normal stress coefficient is considered, which is defined as (3.39) and becomes (3.40) in the linear steady-state regime (see [3.1]). The factor 2 follows from Lodge’s theory of elastic liquids [3.3]. As can be seen from Equation 3.40, the normal stress coefficient in the linear range is a function of the linear steady-state recoverable compliance and the zero-shear viscosity . The elastic compliance , which is of primary interest when elastic properties are discussed, can be obtained by (3.41) with
.
15
16
3 Principles of the Determination of Elastic Properties
In principle, from the normal stress and the shear stress measurements performed in parallel in the linear and steady-state regime of a stressing experiment, the elastic compliance as a characteristic quantity of the elastic properties of a melt can be determined. From Equation 3.40 it becomes obvious that normal stress differences alone may be used as a comparative qualitative measure of elasticity only for samples that have comparable viscosities, even in the range of linear viscoelasticity.
3.5 Capillary Rheometry The elastic properties of a polymer melt become obvious from the swell of a strand extruded through a die, as sketched in Figure 3.4. The shape of the extrudate depends on the geometry of the die and, therefore, is generally not easy to quantify. For studying the extrudate swell, circular capillaries are applied as shown in Figure 3.4. The measured quantity is the diameter of the exiting strand . The extrudate swell is defined as (3.42) or (3.43) with
being the diameter of the capillary.
The definition of Equation 3.43 is preferentially used in this book, because it directly describes the swell by the relative increase of the diameter and is physically is often found for the extrumore descriptive. In the literature the definition date swell. The extrudate swell is a very obvious manifestation of the elastic properties of a melt, but its interpretation is complex as sketched in Section 2.2. Furthermore, some care has to be taken to get well-defined reliable data, because the strand may be frozen before the equilibrium of the recovery is reached (see Section 4.2.5). Moreover, the linear regime is difficult to attain due to the pressures and shear stresses, respectively, necessary for reliable measurements. This fact and the dependence of the extrudate swell on geometrical features of the die and the flow fields in its entry region make it necessary to be aware of the complexity of the extrudate swell as a measure of elastic properties. Nevertheless, the extrudate swell offers some insight into the role of elasticity for processing, as described in Chapter 12.
3.5 Capillary Rheometry
p
d0 L
d
Figure 3.4 Schematics of the swell of a polymer melt extruded by the pressure through a and the length . is the diameter of the exiting strand die with the radius
An effect in capillaries that is due to elastic properties of polymer melts is the so. It occurs in capillaries, when one of two pressure called hole pressure transducers facing each other is not flush-mounted [3.4]. is the pressure measured by a gauge flush-mounted to the wall of a slit capillary and the pressure at the bottom of a hole in the wall just opposite the position of the other transducer. Detailed calculations of various flow patterns around holes are given in [3.5]. For the hole shaped as a slot perpendicular to the flow direction, from measurements of the hole pressure as a function of the shear stress at the wall, the first normal follows (see [3.6]): stress difference (3.44) In principle, this method makes it possible to measure normal stress differences at shear stresses higher than those in rotational rheometry and thus closer to processing conditions. But the experimental difficulties, particularly with the pressure gauges in the case of polymer melts, have hindered a widespread practical use so far.
17
18
3 Principles of the Determination of Elastic Properties
3.6 Recoverable Elongation A very obvious effect of an elastic deformation is the shape recovery of a uniaxially stretched sample after setting the previously applied tensile stress to zero. In the case of a metal spring the total elongation recovers spontaneously; a cross-linked rubber shows a complete recovery too, but with some time lag; and a polymer melt may only exhibit a time-dependent partial rebound of the extended sample. In the Cauchy measure the strain exerted on a sample in a uniaxial tensile experiment is (3.45) and the recoverable strain follows as (3.46) where is the length of the stretched sample and the initial sample length. is the length of the retarded sample, which is a function of time for viscoelastic materials and reaches an equilibrium value after long enough retardation times. For larger deformations, as in the case of rubber or polymer melts, the Hencky strain (3.47) is commonly used and the corresponding recoverable portion reads (3.48) is the total stretching ratio and
the recoverable stretching ratio.
Two limiting cases may be discussed to elucidate Equation 3.48, which is not so common in rheology. When the length after retardation is the same as that of the elongated sample , then evidently the recoverable elongation is zero. Attaining the length of the initial sample after retardation obviously means that the reversible deformation is equal to the total elongation. The strains in the Cauchy and Hencky measures approach each other when the total or recoverable deformations of a sample are small. For the characterization of material-specific properties the compliances are used. In the case of the Hencky measure used for polymer melts, they are defined as (3.49)
3.7 References
for the compliance and (3.50) for the recoverable compliance in elongation, with being the tensile stress. For the matter of convenience, the index H is omitted for the compliances determined in elongation. The steady-state elastic compliance is defined as (3.51) and becomes in the linear regime of deformation. Using the Cauchy measure for the determination of recoverable tensile compliances leads to values different from those in the Hencky measure. They become comparable for small recoverable strains, when the Hencky strain approaches the Cauchy strain. and
are related to each other by (3.52)
3.7 References [3.1]
Münstedt H., Schwarzl F. R., Deformation and Flow of Polymeric Materials, Springer, Berlin (2014)
[3.2]
Macosko C. W., Rheology: Principles, Measurements, and Applications, Wiley-VCH, New York (1994)
[3.3]
Lodge A. S., Elastic Liquids, Academic Press, New York (1964)
[3.4]
Higashitani K., Lodge A. S., Hole pressure error measurements in pressure-generated shear flow, Trans. Soc. Rheol. 16 (1972), 307–335
[3.5]
Higashitani K., Pritchard W. G., A kinematic calculation of intrinsic errors in pressure measurements made with holes, Trans. Soc. Rheol. 16 (1972), 687–696
[3.6]
Ferry J. D., Viscoelastic Properties of Polymers, John Wiley, New York (1980)
19