**2. Theoretical background**

### **2.1 Stress relaxation and determination of relaxation spectra**

Polymers exposed to constant strain exhibit the well known phenomenon of stress relaxation, i.e. a more or less strong decrease of stress as a function of time. The microscopic

Characterization of Thermoplastic Elastomers

calculated in first approximation by Eq. (3).

perature scan of a TSSR test.

those effects will be described in detail.

**2.2 Thermal expansion of the sample** 

behaviour.

by Means of Temperature Scanning Stress Relaxation Measurements 349

All relaxation mechanisms, and thus the relaxation time constants, are strongly depending on temperature, i.e. the higher the temperature, the lower the relaxation time constants and vice versa. Therefore, the relaxation spectrum covers a very wide range on time scale and it is practically impossible to determine the entire function by means of a single stress relaxation measurement. Usually, a set of measurements at several temperatures have to be executed, to create a master curve, based on time - temperature superposition principle (TTS). That means, high effort is required to obtain full information of the stress relaxation

With temperature scanning stress relaxation (TSSR) measurements, an alternative strategy has been introduced recently (Vennemann et al., 2001; Vennemann, 2003; Barbe et al., 2005). In contrast to traditional isothermal tests, during TSSR measurements the temperature is not kept constant, but is increasing linearly with a constant heating rate β, e.g. β = 2 K/min. As a result, the non-isothermal relaxation modulus Enon-iso as a function of temperature is obtained. Analogue to isothermal stress relaxation measurements, the spectrum H(T) can be

( ) *non iso*

*dT*

with *TT T t* <sup>0</sup>

In this equation T0 stands for the initial temperature at which the test is started and β is the heating rate of the temperature scan. Although this function is not defined on time scale, the relaxation mechanisms the polymer sample undergoes during the test can be identified, clearly, because the relaxation time constant decreases monotonously with increasing temperature T. Due to its very strong temperature dependence, the relaxation time constant drops down to small values rapidly, within small temperature range. Thus, the entire spectrum is observable on temperature scale within a relative short period of time during a tem-

Beside stress relaxation, two other phenomena, i.e. thermal expansion and rubber elasticity of the sample, have to be taken into consideration, if a sample, mounted between two sample holders having constant distance, is heated up linearly. In the following sections,

Due to thermal expansion of the material the initial length l0 of the sample is increasing, if a temperature scan is performed starting at initial temperature T0 up to higher temperatures T. In consequence, the thermal expansion of the sample contributes to a decrease of stress, if a stretched sample is mounted between sample holders with constant distance. The thermally

induced variation of strain = (l - l0)/l0 during TSSR tests can be easily calculated by

*dE HT T*

*dE dE H t*

*iso iso t t*

*<sup>T</sup> const <sup>t</sup>*

(3)

(2)

*d t dt* 

'( ) ln

mechanisms, leading to the macroscopic recognizable decrease of stress, may result from physical and/or chemical processes. In contrast to thermoset rubber, where the thermal mechanical behaviour is dominated by chemical reactions resulting in cleavage of polymer chains and network junctions, in case of thermoplastic elastomers physically induced stress relaxation processes are most important with respect to usage properties.

For the simple Maxwell - model, as represented in Figure 1, the relaxation time constant is defined as the period of time, after the stress has dropped down to the value of 0/e. Here, 0 indicates the initial stress at time zero when the strain has been applied to the sample. Unfortunately, the real behaviour of materials is more complicated and cannot be described by the simple Maxwell - model. According to the well known theory of linear viscoelasticity the entire relaxation process can be described by means of the generalized Maxwell - model, which consists of an infinite number of individual spring - dashpot - elements. Under isothermal conditions (T = const.) the relaxation modulus Eiso is a function of time t and given by Eq. (1), for this model (Ferry, 1980).

$$E\_{iso}(t) = E\_{\infty} + \int\_{-\infty}^{\infty} H'(\tau) \cdot e^{\frac{-t}{\tau}} \, d\ln \tau \tag{1}$$

Fig. 1. Maxwell - Model and stress as a function of time after a constant strain 0 has been applied at t = 0.

The relaxation modulus is also directly related to the experimentally observable stress (t) and can be easily calculated by dividing the stress by the applied strain 0. In this equation the relaxation spectrum H'() is a steady function describing the probability of the relaxation time constants of the model which may be associated with the population of relaxation mechanisms of the system. The constant E is added in Eq. (1) to allow the system to approach an equilibrium state higher than zero, as observed normally for viscoelastic solids.

According to Alfrey's rule (Alfrey & Doty, 1945) the value of the relaxation spectrum H'() at point = t is obtained in first approximation by differentiating *Eiso(t)* with respect to ln*t,*  by Eq.(2).

mechanisms, leading to the macroscopic recognizable decrease of stress, may result from physical and/or chemical processes. In contrast to thermoset rubber, where the thermal mechanical behaviour is dominated by chemical reactions resulting in cleavage of polymer chains and network junctions, in case of thermoplastic elastomers physically induced stress

For the simple Maxwell - model, as represented in Figure 1, the relaxation time constant is defined as the period of time, after the stress has dropped down to the value of 0/e. Here, 0 indicates the initial stress at time zero when the strain has been applied to the sample. Unfortunately, the real behaviour of materials is more complicated and cannot be described by the simple Maxwell - model. According to the well known theory of linear viscoelasticity the entire relaxation process can be described by means of the generalized Maxwell - model, which consists of an infinite number of individual spring - dashpot - elements. Under isothermal conditions (T = const.) the relaxation modulus Eiso is a function of time t and given

( ) '( ) ln

*E t E H ed iso*

0

Fig. 1. Maxwell - Model and stress as a function of time after a constant strain 0 has been

The relaxation modulus is also directly related to the experimentally observable stress (t) and can be easily calculated by dividing the stress by the applied strain 0. In this equation the relaxation spectrum H'() is a steady function describing the probability of the relaxation time constants of the model which may be associated with the population of relaxation mechanisms of the system. The constant E is added in Eq. (1) to allow the system to approach an equilibrium state higher than zero, as observed normally for viscoelastic solids. According to Alfrey's rule (Alfrey & Doty, 1945) the value of the relaxation spectrum H'() at point = t is obtained in first approximation by differentiating *Eiso(t)* with respect to ln*t,* 

 

stress

0/e

σ

0

*t*

time t

(1)

strain 0 = const temperature T = const

 

relaxation processes are most important with respect to usage properties.

by Eq. (1), for this model (Ferry, 1980).

applied at t = 0.

by Eq.(2).

$$H'(\tau) = -\left(\frac{dE\_{iso}}{d\ln t}\right)\_{t=\tau} = -t \cdot \left(\frac{dE\_{iso}}{dt}\right)\_{t=\tau} \tag{2}$$

All relaxation mechanisms, and thus the relaxation time constants, are strongly depending on temperature, i.e. the higher the temperature, the lower the relaxation time constants and vice versa. Therefore, the relaxation spectrum covers a very wide range on time scale and it is practically impossible to determine the entire function by means of a single stress relaxation measurement. Usually, a set of measurements at several temperatures have to be executed, to create a master curve, based on time - temperature superposition principle (TTS). That means, high effort is required to obtain full information of the stress relaxation behaviour.

With temperature scanning stress relaxation (TSSR) measurements, an alternative strategy has been introduced recently (Vennemann et al., 2001; Vennemann, 2003; Barbe et al., 2005). In contrast to traditional isothermal tests, during TSSR measurements the temperature is not kept constant, but is increasing linearly with a constant heating rate β, e.g. β = 2 K/min. As a result, the non-isothermal relaxation modulus Enon-iso as a function of temperature is obtained. Analogue to isothermal stress relaxation measurements, the spectrum H(T) can be calculated in first approximation by Eq. (3).

$$H(T) = -\Delta T \cdot \left(\frac{dE\_{nom-iso}}{dT}\right)\_{\beta = \frac{\Delta T}{t} = const} \tag{3}$$

$$\text{with } T - T\_0 = \Delta T = \mathcal{J} \cdot t$$

In this equation T0 stands for the initial temperature at which the test is started and β is the heating rate of the temperature scan. Although this function is not defined on time scale, the relaxation mechanisms the polymer sample undergoes during the test can be identified, clearly, because the relaxation time constant decreases monotonously with increasing temperature T. Due to its very strong temperature dependence, the relaxation time constant drops down to small values rapidly, within small temperature range. Thus, the entire spectrum is observable on temperature scale within a relative short period of time during a temperature scan of a TSSR test.

Beside stress relaxation, two other phenomena, i.e. thermal expansion and rubber elasticity of the sample, have to be taken into consideration, if a sample, mounted between two sample holders having constant distance, is heated up linearly. In the following sections, those effects will be described in detail.

#### **2.2 Thermal expansion of the sample**

Due to thermal expansion of the material the initial length l0 of the sample is increasing, if a temperature scan is performed starting at initial temperature T0 up to higher temperatures T. In consequence, the thermal expansion of the sample contributes to a decrease of stress, if a stretched sample is mounted between sample holders with constant distance. The thermally induced variation of strain = (l - l0)/l0 during TSSR tests can be easily calculated by

Characterization of Thermoplastic Elastomers

described by Eq. (4), the relation of Eq.(5) can be rewritten as

 

 

black filled elastomers (Vennemann et al., 2011).

**2.4 General remarks** 

*R T*

al value of at temperature T0 is given by Eq. (8) (Vennemann et al., 2011).

 

   

by Means of Temperature Scanning Stress Relaxation Measurements 351

 <sup>2</sup> *T R* 

 

0 0

*T T*

 

 

2 2

 

( ) *TR T* <sup>2</sup> (8)

1 1

where 0 is the initial strain ratio at temperature T0. The influence of thermal expansion on the stress - temperature - curve is shown in Figure 3, where the uncorrected curve as calculated from Eq. (5), is represented in comparison to the corrected curve, as calculated from Eq. (7). Obviously, the initial slope is slightly reduced due to thermal expansion. Furthermore, the corrected function is no longer strictly linear, but exhibits a slight curvature with increasing temperature. From Eq. (7), by derivation with respect to temperature T, the corrected temperature coefficient is obtained, which is now also a function of temperature. The initi-

0 0 00 0 0 0

It has been shown; macroscopic recognizable increase of stress is recognizable if an elongated piece of rubber is heated up linearly. The macroscopic reaction of the material is caused by the change of entropy on microscopic scale and thus, it becomes possible to easily determine the crosslink density of a rubber sample, which is an important microscopic parameter of the system. But, it is important to notice, that the above equations are only strictly correct for ideal rubber networks. Real systems, such as filled elastomers and thermoplastic elastomers, are more complex, and cannot be fully described by this simple theory. Therefore, further development of theory is required to better understand the behaviour of those materials. Recently, a model has been developed to describe the thermoelastic behaviour of carbon

In case of thermoplastic elastomers the situation is even more difficult, because these materials consist of at least two phases and in case of commercial grades additionally of fillers, plasticizer and other additives. Although most elastomers and also thermoplastic elastomers (TPE) exhibit thermoelastic behaviour similar to ideal rubber, calculation of true crosslink density is not possible, but only apparent values because of lack of adequate theory. Nevertheless, the characterization of thermoelastic behaviour by TSSR measurements is very useful, in particular in case of thermoplastic vulcanizates, because properties which are closely related to the structure of the polymer network become recognizable. Furthermore, additio-

(6)

2

 

(7)

 

For high elongations, the temperature coefficient is positive, however at low strain ratios, i.e. < 1.1, a negative value of was found experimentally. The transition from a negative to a positive value of the temperature coefficient is known as thermoelastic inversion (Pellicer, 2001) . The phenomenon of thermoelastic inversion is not predicted by theory but, it was shown early (Anthony et al., 1942) and has been confirmed by own measurements (Vennemann & Heinz, 2008), this apparent contradiction results only from thermal expansion of the sample. Considering the temperature dependence of the strain, as

$$
\varepsilon(T) = \frac{1}{L\_0(1 + \alpha \cdot \Delta T)} - 1 \tag{4}
$$

where L0 is the initial distance of the sample holders at temperature T0 and is the coefficient of linear thermal expansion of the sample. For rubber typical values for are in the range of 1 to 3 . 10-4 K-1 (Gent, 2001). In Figure 2 the relative strain /0 is plotted against temperature for certain values of initial strain 0. It becomes obvious, increasing temperature results in decreasing strain. However, the influence of temperature on relative strain is more or less negligible, if the strain is sufficiently high. To minimize the influence of thermal expansion, TSSR experiments should be performed at initial strains not below 50 %.

Fig. 2. Influence of thermal expansion of the sample on relative strain. Curves were calculated for certain initial strain values, as indicated, by means of Eq. (4) with = 3 . 10-4 K-1.

#### **2.3 Rubber elasticity**

According to the well known theory of rubber elasticity in case of an ideal rubber network, the mechanical stress is proportional to the absolute temperature T and can be expressed by Eq. (5) (Mark, 1981).

$$
\sigma = \nu \cdot \mathbb{R} \cdot T \left( \mathcal{X} - \mathcal{X}^{-2} \right) \tag{5}
$$

Where ν is the crosslink density of the network and R is the universal gas constant. The strain ratio is defined as = l/l0, where l is the length and l0 the initial length of the sample. According to Eq. (5) the stress should increase with increasing temperature, if the strain ratio is kept constant. The slope of the stress - temperature - curve at constant elongation can be obtained from the derivative of stress with respect to temperature which is assigned as temperature coefficient in the following.

$$\boldsymbol{\kappa} = \left(\boldsymbol{\hat{\boldsymbol{\sigma}}} \boldsymbol{\sigma} / \boldsymbol{\hat{\boldsymbol{\sigma}}} \boldsymbol{\Gamma}\right)\_{\boldsymbol{\lambda}} = \boldsymbol{\nu} \cdot \boldsymbol{\mathcal{R}} \cdot \left(\boldsymbol{\lambda} - \boldsymbol{\lambda}^{-2}\right) \tag{6}$$

For high elongations, the temperature coefficient is positive, however at low strain ratios, i.e. < 1.1, a negative value of was found experimentally. The transition from a negative to a positive value of the temperature coefficient is known as thermoelastic inversion (Pellicer, 2001) . The phenomenon of thermoelastic inversion is not predicted by theory but, it was shown early (Anthony et al., 1942) and has been confirmed by own measurements (Vennemann & Heinz, 2008), this apparent contradiction results only from thermal expansion of the sample. Considering the temperature dependence of the strain, as described by Eq. (4), the relation of Eq.(5) can be rewritten as

$$
\sigma = \nu \cdot R \cdot T \left( \frac{\lambda\_0}{1 + a \cdot \Delta T} - \left( \frac{\lambda\_0}{1 + a \cdot \Delta T} \right)^{-2} \right) \tag{7}
$$

where 0 is the initial strain ratio at temperature T0. The influence of thermal expansion on the stress - temperature - curve is shown in Figure 3, where the uncorrected curve as calculated from Eq. (5), is represented in comparison to the corrected curve, as calculated from Eq. (7). Obviously, the initial slope is slightly reduced due to thermal expansion. Furthermore, the corrected function is no longer strictly linear, but exhibits a slight curvature with increasing temperature. From Eq. (7), by derivation with respect to temperature T, the corrected temperature coefficient is obtained, which is now also a function of temperature. The initial value of at temperature T0 is given by Eq. (8) (Vennemann et al., 2011).

$$\kappa\_0 = \kappa(T\_0) = \nu \cdot R \cdot \left[ \left( \lambda\_0 - \lambda\_0^{-2} \right) - T\_0 \cdot \alpha \cdot \left( \lambda\_0 + 2 \cdot \lambda\_0^{-2} \right) \right] \tag{8}$$

#### **2.4 General remarks**

350 Thermoplastic Elastomers

*L T*

where L0 is the initial distance of the sample holders at temperature T0 and is the coefficient of linear thermal expansion of the sample. For rubber typical values for are in the range of 1 to 3 . 10-4 K-1 (Gent, 2001). In Figure 2 the relative strain /0 is plotted against temperature for certain values of initial strain 0. It becomes obvious, increasing temperature results in decreasing strain. However, the influence of temperature on relative strain is more or less negligible, if the strain is sufficiently high. To minimize the influence of thermal

0 50 100 150 200

<sup>2</sup>

*R T* (5)

 

Where ν is the crosslink density of the network and R is the universal gas constant. The strain ratio is defined as = l/l0, where l is the length and l0 the initial length of the sample. According to Eq. (5) the stress should increase with increasing temperature, if the strain ratio is kept constant. The slope of the stress - temperature - curve at constant elongation can be obtained from the derivative of stress with respect to temperature which is assigned

temperature T / °C

Fig. 2. Influence of thermal expansion of the sample on relative strain. Curves were calculated for certain initial strain values, as indicated, by means of Eq. (4) with = 3 . 10-4 K-1.

According to the well known theory of rubber elasticity in case of an ideal rubber network, the mechanical stress is proportional to the absolute temperature T and can be expressed

(4)

10% 20% 50% 100%

0 ( ) 1 (1 )

*<sup>l</sup> <sup>T</sup>*

expansion, TSSR experiments should be performed at initial strains not below 50 %.

0

0.2

0.4

0.6

relative strain ε/ε0

**2.3 Rubber elasticity** 

by Eq. (5) (Mark, 1981).

as temperature coefficient in the following.

0.8

1

1.2

It has been shown; macroscopic recognizable increase of stress is recognizable if an elongated piece of rubber is heated up linearly. The macroscopic reaction of the material is caused by the change of entropy on microscopic scale and thus, it becomes possible to easily determine the crosslink density of a rubber sample, which is an important microscopic parameter of the system. But, it is important to notice, that the above equations are only strictly correct for ideal rubber networks. Real systems, such as filled elastomers and thermoplastic elastomers, are more complex, and cannot be fully described by this simple theory. Therefore, further development of theory is required to better understand the behaviour of those materials. Recently, a model has been developed to describe the thermoelastic behaviour of carbon black filled elastomers (Vennemann et al., 2011).

In case of thermoplastic elastomers the situation is even more difficult, because these materials consist of at least two phases and in case of commercial grades additionally of fillers, plasticizer and other additives. Although most elastomers and also thermoplastic elastomers (TPE) exhibit thermoelastic behaviour similar to ideal rubber, calculation of true crosslink density is not possible, but only apparent values because of lack of adequate theory. Nevertheless, the characterization of thermoelastic behaviour by TSSR measurements is very useful, in particular in case of thermoplastic vulcanizates, because properties which are closely related to the structure of the polymer network become recognizable. Furthermore, additio-

Characterization of Thermoplastic Elastomers

a. Commercial grades based on EPDM/PP

by Means of Temperature Scanning Stress Relaxation Measurements 353

Several commercial grades of thermoplastic vulcanizates based on EPDM/PP covering a wide range of hardness were obtained from Solvay Engineered Polymers (TX/USA) and tested as received. The novel TPV-AP materials were produced via a dynamic vulcanization process using a new curative system and DVA process developed by Solvay Engineered Polymers (Reid et al., 2004). The new cure system results in a material with non-hygroscopic behaviour, white colour, and low odour. Properties of TPV-AP are compared to two other commercially available TPV materials. TPV-HS is a commercial TPV based upon EPDM and PP where the elastomer is crosslinked with a hydrosilation process. TPV-PH is also a commercial TPV based upon EPDM and PP where the elastomer is crosslinked with a phenolic resin curing process. The samples of both TPV-HS and TPV-PH were not produced by

**3.1.2 Thermoplastic polyolefin blends (TPO) and dynamic vulcanizates (TPV)** 

Solvay Engineered Polymers, but commercial grades, produced by other suppliers.

Commercial available EPDM rubber and isotactic polypropylene homopolymer (PP) were used as the basis for the dynamic vulcanizates (TPV). The EPDM contains 50 wt % ethylene and 4 wt % ethylidene norbornene (ENB). It has a Mooney viscosity ML(1+4) at 125 °C, of 36. The melt flow rate of the polypropylene, measured at 230 °C and 2.16 kg is 12 g/10 min. The crosslink system consists of di(tert-butylperoxyisopropyl)benzene (abbrev.: DTBPIB) as peroxide and trimethylolpropane trimethacrylate (abbrev.: TRIM) as co-agent. The peroxide and co-agent are supplied commercially on a silica carrier, with active agent content of 40 wt % and 70 wt %, respectively. The TPV samples are designated as TPV1 to TPV6, whereas the total amount of curatives (DTBPIB and TRIM) is increasing from 1 phr to 6 phr in steps of 1 phr. The volume fraction of polypropylene was PP = 0.23 in all compounds. An uncured compound of identical EPDM/PP ratio was also produced and tested as reference sample. All samples were produced in a two-step mixing process using a Haake Rheocord 600 laboratory internal mixer (Thermo Electron Corporation, Karlsruhe). Further details of

Two different commercially available EPDM rubber and two different grades of high density polyethylene (HDPE) were used as the basis for the dynamic vulcanizates, in this study. Crosslinking of the EPDM in all compounds was performed with a phenolic resin cure system, consisting of stannous chloride (SnCl2 2 H2O), zinc oxide (ZnO) and SMD 31214. The latter is a commercially available solution of paraffinic mineral oil and 30 wt % of phenolic resin SP 1045. Further details of the composition and preparation of the compounds

The temperature scanning stress relaxation tests were performed by use of a commercial available TSSR instrument obtained from Brabender GmbH (Duisburg, Germany). The TSSR instrument (Fig. 4) consists of an electrical heating chamber where the sample, a S2 testing rod, is placed between two clamps. The clamps are connected to a linear drive unit to apply

b. Model compounds of peroxide cured TPV based on EPDM/PP

the production process are published elsewhere (Vennemann, 2006). c. Model compounds of phenolic cured TPV based on EPDM/HDPE

are published elsewhere (Vennemann, 2009).

**3.2 TSSR instrument and test procedure** 

nal information about the composition, morphology and structure of the sample can be deduced from the entire relaxation spectrum.

Fig. 3. Theoretical stress - temperature - curves, calculated by use of Eq. (5) and Eq. (7) with ν = 100 mol/m3, 0 = 1.5 and = 3 . 10-4 K-1 .

#### **3. Experimental**

#### **3.1 Materials and preparation of the samples**

#### **3.1.1 Thermoplastic elastomers based on Styrene Block Copolymers (SBC)**

High molecular weight poly(styrene-b-ethylene/butylene-b-styrene) (SEBS) with a polystyrene (PS) content of 33%, a molar mass of the PS-blocks of 29000 g/mol and a total molar mass of *M*w = 174000 g/mol were used as the basis for the compounds prepared. In SBC/ polyolefin blends a standard isotactic polypropylene and, alternatively a standard high density polyethylene were used as the polyolefin component of the compounds. In SBC/PPE blends high molecular weight poly(p-phenylene ether) (PPE) with Tg = 215°C and molar mass of Mw = 38900 g/mol was used as the modifier. Additionally, high purity medicinal paraffin oil was used as the extender oil for all compounds and a small amount of stabilizer was added to protect the polymers against degradation during the mixing process.

SBC/polyolefin compounds were produced using a twin-screw extruder (L/D: 32/1, 25 mm diameter; Berstoff GmbH). SBC/PPE compounds were produced by means of a single-screw extruder (Göttfert GmbH, L/D: 15/1). In all cases the ingredients were mixed together prior feeding to the extruder, having a barrel temperature of 260°C. Test plates of 2mm thickness of all compounds were produced in a pneumatic injection moulding press. Further details are described in earlier papers (Vennemann et al., 2004) and (Barbe et al., 2005).
