Rheologica Acta

, Volume 45, Issue 5, pp 717–727 | Cite as

Rheological properties of branched polystyrenes: nonlinear shear and extensional behavior

Original contribution


Nonlinear shear and uniaxial extensional measurements on a series of graft-polystyrenes with varying average numbers and molar masses of grafted side chains are presented. Step-strain measurements are performed to evaluate the damping functions of the melts in shear. The damping functions show a decreasing dependence on strain with an increase in mass fraction of grafted side chains. Extensional viscosities of the melts of graft-polystyrenes exhibit a growing strain hardening with increasing average number of grafted side chains as long as the side branches have a sufficient molar mass to be entangled. Graft-polystyrenes with side arms below the critical molar mass Mc for entanglements of linear polystyrene but above the entanglement molar mass Me of linear polystyrene (Me < Mw,br < Mc) still show a distinct strain hardening. With decreasing molar mass of the grafted side chains (Mw,br < Me) the nonlinear-viscoelastic properties of the graft-polystyrene melts approach the behavior for a linear polystyrene with comparable polydispersity.


Graft-polystyrenes Long-chain branching Shear and elongational viscosity Damping function in shear and elongation Elongational viscosity Strain hardening Entanglements 


Many nonlinear rheological properties of polymer melts are influenced by long-chain branches. For example, an increased shear thinning behavior compared to the properties of the corresponding linear polymers is found for long-chain branched polyethylenes, polystyrenes and polycarbonates (e.g., Heino et al. 1992; Gabriel and Münstedt 1999; Ferri and Lomellini 1999; Grigo et al. 1996).

Another important nonlinear feature is the strain-hardening behavior in uniaxial extensional flow. Some chemically branched or electron beam irradiated polypropylenes show an increased strain hardening when compared to their linear counterparts (Hingmann and Marczinke 1994; Kurzbeck et al. 1999; Gotsis et al. 2004). For polyethylene, it is well known that long-chain branched LDPE exhibits a pronounced strain hardening in experiments with constant strain rate. Experiments at constant stresses have shown that the maximum of the steady-state elongational viscosity increases with the degree of branching (Münstedt and Laun 1981). However, the authors pointed out that there is a lack of reliable characterization methods for these long-chain branched polymers. Moreover, the molar mass distribution of the higher branched LDPE studied by Münstedt and Laun (1981) is broader compared to the one with a lower branching content, which makes an unambigous interpretation of the influence of branching on strain hardening more difficult.

The strain-hardening due to long-chain branching has important implications for processing operations, and particularly for those in which free surface flow occurs, e.g., in film blowing, blow molding, thermoforming or foaming. As an example, a better homogeneity of the thickness of blown films from low-density polyethylene (LDPE) compared to that one for a nonstrain hardening linear-low density polyethylene (LLDPE) has been found (Kurzbeck 1999).

To understand the influence of long-chain branching on rheological properties, melts or solutions of model polymers like stars and H-type structures have been studied mainly in linear viscoelastic flow. Only a very few experimental studies of nonlinear-viscoelastic rheological properties on well-defined star- or H-type polymers are reported (Pearson et al. 1983; McLeish et al. 1999). Detailed studies on nonlinear-viscoelastic properties of well-defined graft-polymers in shear and in elongational flow are not available from literature. Therefore, from the rheological experiments on grafted polystyrenes in the nonlinear regime reported in this paper, new insights into the role of long-chain branches can be expected.

The linear-viscoelastic properties of the materials studied here are discussed in another paper (Hepperle et al. 2005). To study the influence of molecular topology of graft-polystyrenes on nonlinear-viscoelastic properties, two experimental methods were chosen:

  1. (1)

    Stress relaxation after shear steps to obtain the time and strain-dependent modulus G(t,γ) and the damping function h(γ).

  2. (2)

    Deformation at a constant Hencky-strain rate \(\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon}_{0} \) in uniaxial elongation to determine the tensile stress \(\sigma ^{+}_{\rm E} (t,\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon})\) and the tensile growth coefficient \(\eta _{\rm E}^{+} (t,\;\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon}) =\sigma ^{+}_{\rm E} (t,\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon})/\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon}.\)


For these two types of flow, the influence of the molecular topology on the material functions is studied. From the tensile stress growth coefficients in uniaxial extension, the damping functions hu(ε) are calculated according to Wagner (1979). The dependence of h(γ) and hu(ε) on molecular structure is compared.



Graft-polystyrene (PS) polymers with a brush-like architecture were synthesized by the macromonomer technique (Haug 2000). The constitution of the sample is presented using the nomenclature \({\text{PS}}{\text{-}}x{\text{-}}\ifmmode\expandafter\bar\else\expandafter\=\fi{p}G{\text{-}}y\) with x describing the number average molar mass Mn,bb in kg/mol of the polydisperse backbone, \(\ifmmode\expandafter\bar\else\expandafter\=\fi{p}\) the average number of grafted side chains per molecule randomly attached to the backbone and y the number average molar mass Mn,br of the side chains in kg/mol. The number average mass fraction Φn,br of grafted side chains is obtained by
$$ \Phi _{{\text{n,br}}} = \frac{{{\bar p}\cdot{M_{{\text{n,br}}} }}} {{M_{{\text{n,bb}}} + {\bar p}\cdot{M_{{\text{n,br}}} }}} $$
values of Φn,br are given in Table 1. The detailed molecular data together with the linear-viscoelastic properties of the samples are given by Hepperle et al. (2005).

Shear rheometry

Step-strain experiments to determine the transient moduli G(t,γ) of the polymers were carried out using an ARES rheometer (Rheometric Scientific) with a cone-plate geometry (plate diameter d=25 mm, cone angle α=5.73°). The sample preparation is described elsewhere (Hepperle et al. 2005). The temperatures applied were chosen in a way that the zero shear viscosities range between 0.7×103 and 13.4×103 Pa (see Table 1), a viscosity level that is sufficiently high to give a stress signal with low enough noise at small strains and that allows measurements up to strains of γ=4.95 without an overload of the force transducer. Step strains ranging from γ=0.01 up to γ=4.95 were imposed. The response time was t<0.1 s for all strains. Before imposing the next step strain, the sample was allowed to relax completely, as indicated by a diminishing torque. The sample was checked visually to ensure that no melt poured out of the gap during experiments at large deformations up to γ=4.95.
Table 1

Data on sample composition and experimental conditions chosen. Sample constitution is described as \({\text {PS}}{\text {-}}x{\text {-}}\ifmmode\expandafter\bar\else\expandafter\=\fi{p}G{\text{-}}y\) with x number average molecular mass Mn,bb (kg/mol) of the backbone, \(\ifmmode\expandafter\bar\else\expandafter\=\fi{p}:\) average number of grafted side chains per molecule, y number average molar mass Mn,br (kg/mol) of the side chains. wMM denotes the mass fraction of low-molecular weight residue, Φ n,br denotes the number average mass fraction of grafted side chains



Φ n,br

Step- Strain

Damping function parameter

Uniaxial elongation

T (°C)

η0 (T)a (Pa s)

T (°C)

η0 (T)b (Pa s)









































































































































aZero shear viscosity at measuring temperature of step-strain experiments, calculated with WLF-coefficients (Hepperle et al. 2005)

bZero shear viscosity at measuring temperature of uniaxial elongational flow experiments, calculated with WLF-coefficients (Hepperle et al. 2005)

Extensional rheometry

The samples were prepared by pressing the powder under vacuum in a cylinder-shaped mold followed by an extrusion at T=170°C through a capillary using a melt flow indexer. The extruded rod was annealed in an oil bath at a temperature adjusted to the measuring temperature of the extensional tests (see Table 1). The rod was then cut into cylinder-shaped samples with lengths of 20–25 mm and diameters of 4–5 mm. The samples were fixed with a glue to aluminum clamps.

The elongational experiments were carried out using the commercial elongational rheometer RER 9000 (Rheometrics). Its principal construction follows the design published by Münstedt (1979). The sample is stretched vertically in a silicone oil bath. The density of the oil bath matches the density of the sample at the measuring temperature as closely as possible. Measuring temperatures from 147.5 to 170°C were used. The tensile force is measured by a force transducer which is submerged within the oil bath. With a maximum displacement of the pull rod of the elongational rheometer of 500 mm and a typical initial sample length L0 between 20 and 25 mm, stretching ratios of λ=20–25 can be achieved which correspond to Hencky-strains of ε=3–3.2 according to \(\varepsilon = \ln \lambda = \ln \frac{L}{{L_{0}}}.\)L is the sample length after elongation.

Results and discussion

Step-strain in shear

Step-strain experiments are widely used for the investigation of nonlinear-viscoelastic properties of polymer melts. From measurements of such kind new insights into effects of different branching topologies of the graft-polystyrenes on the nonlinear shear behavior are expected.

For many polymer melts and solutions, the relaxation modulus G(t,γ)=σ(t,γ)/γ is experimentally found to be separable into a strain and time-dependent term above a certain critical time tc according to the equation
$$G (t,\gamma) = h(\gamma)\times G(t), \quad t>t_{c}.$$

The time-dependent function G(t) is the linear relaxation modulus, h(γ) is called the damping function.

Time-strain separability

The relaxation moduli G(t,γ) for the linear, radically synthesized polystyrene PS-r-95 are shown in Fig. 1. The delay in imposition of strain as well as the compliance of the torque transducer effect that correct values of G(t,γ) can be expected for times t>0.15 s (Fig. 1). For long times, i.e., small torques the evaluation is limited by the noise of the signals. G(t,γ) can therefore only be evaluated in a certain time window, as indicated in Fig. 1 for PS-r-95 at 180°C.
Fig. 1

Deformation γ(t) and relaxation modulus G(t,γ) for the melt of the linear polystyrene PS-r-95 at the strains γ0=0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 0.8, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5 and 4.95

For the graft-polystyrenes, the shear relaxation moduli G(t,γ) measured by step-shear strains of magnitudes ranging from γ=0.1 to γ=4.95 are shown in Fig. 2. As the curves for each sample in Fig. 2 are parallel to each other they can be shifted along the vertical axis to give a master curve, i.e., the separability is given and h(γ) can experimentally be determined. Values for h(γ) are presented in Figs. 3, 4 and 5.
Fig. 2

Time-dependent nonlinear shear relaxation modulus G(t,γ) at different strains γ=0.1 (dotted line, top), 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 4.95 (dashed line, bottom). Measuring temperatures are indicated in the graphs

Fig. 3

Influence of measuring temperatures (a, b) and influence of low-molecular weight residue (c) on the shear-damping functions h(γ)

Fig. 4

Damping functions h(γ) for different graft-polystyrenes with varying molar masses of grafted side chains. For all graphs: (dotted line) PSM-damping functions (Eq. 3) with parameter α as indicated in Table 1, (solid line) Soskey–Winter damping functions (Eq. 4) with parameters a and b as indicated in Table 1, (lines with dot) approximation of Doi–Edwards function (DE) (Larson 1985). All graphs contain the damping function of PS-r-95 (open squares) for comparison. aGraft-polystyrenes with grafted side chains of Mw,br of 4.54 and 5.83×104 g/mol. bGraft-polystyrenes with grafted side chains of Mw,br of 2.27 and 2.86×104 g/mol. Data for PS-70-3.2-22 and PS-90-1.2-27 are shifted vertically by a factor of 2. Fit of damping functions for PS-r-95 (Eqs. 3, 4) are not included. cGraft-polystyrenes with grafted side chains of Mw br of 0.68 and 1.42×104 g/mol. Fit of damping functions for PS-r-95 (Eqs. 3, 4) are not included

Fig. 5

Influence of the weight fraction Φ n,br of grafted side chains on the damping function in shear for graft-polystyrene melts. Φ n,br= 0 denotes the linear PS. DE approximation of Doi–Edwards function (Larson 1985)

Influence of temperature and low-molar mass residues

As can be seen from Fig. 3a and b on two different samples, the damping function is independent of temperature for the range of temperatures shown. It was also proven that the low-molecular weight residues do not alter the strain dependence (Fig. 3c): The damping functions of PS-60-0.5G-55 and PS-60-0.5G-55* are indistinguishable, as well as the ones of PS-60-2.1G-27 and PS-55-2.1G-27*. Values for the mass fraction of the low-molar mass residues wMM of the samples are given in Table 1.

Influence of branching on the damping function in shear

Figure 4 shows the experimentally determined damping functions for all graft-polystyrene melts investigated, ordered in groups with longer side chains (Fig. 4a), shorter side chains (Fig. 4b) and shortest side chains (Fig. 4c).

For a quantification of the influence of the molar structure on the strain dependence of h(γ), the damping function has to be fitted to an empirical function. Many functions for the damping function have been proposed for both shear and extensional flows. For a description of h(γ) mainly exponential (Wagner 1979) and sigmoidal functions (Soskey and Winter 1984; Papanastasiou et al. 1983) are used. Sigmoidal-shaped functions are preferred as they describe the transition of h(γ) from the linear-viscoelastic to the nonlinear region better than the exponential forms. Two sigmoidal functions are used. The first one,
$$h(\gamma) = \frac{\alpha}{{\alpha + \gamma ^{2}}}$$
is called PSM-function where α is a specific parameter (Papanastasiou et al. 1983).
The other function derived by Soskey and Winter (1984) contains the additional adjustable parameter b as the strain exponent:
$$h(\gamma) = \frac{a}{{a + \gamma ^{b}}}$$

As can be seen from the dotted lines in Fig. 4 representing Eq. 3, the experimental data are slightly overpredicted at low strains and underpredicted at high strains. The full lines in Fig. 4 show the numerical descriptions obtained with Eq. 4. From the comparison it can be concluded that Eq. 4 is more appropriate for a numerical description than Eq. 3. Data for α, a and b are given in Table 1.

For the damping function according to the theory of Doi and Edwards (1986), several simple approximations have been proposed (Larson 1985; Takahashi et al. 1993), of which the one by Larson is applied in this work. It corresponds to Eq. 3 with α=5. This approximation for the Doi–Edwards damping function is marked by DE in the figures.

The fits for PS-r-95 are given in Fig. 4a. The damping function of the linear polystyrene PS-r-95 is always included in the diagrams (open squares). The value of α=7.4 for PS-r-95 is close to the one from previous studies on melts of linear PS, for which values of 5.5 (Mw=398,000 g/mol, Mw/Mn=2.9) and 7.9 (Mw=220,000 g/mol, Mw/Mn=2.44) are given by Khan et al. (1987).

Figure 4a shows two graft-polystyrenes with the same number average molar masses Mn,bb of the backbone (60 kg/mol) and with grafted side chains of the molar masses 42 and 55 kg/mol, which lie above the critical molar mass Mc≅35 kg/mol for entanglements. The two graft-polystyrenes in Fig. 4a have average numbers of 1.4 and 0.5 grafted chains per molecule. PS-60-1.4G-42 shows a damping function which is less dependent on strain than that one of PS-60-0.5G-55 with a lower average number of grafted side chains. These two branched polystyrenes exhibit a weaker damping than the linear polystyrene, PS-r-95, which has a similar polydispersity as the graft-polystyrenes. It can therefore be supposed that at nearly constant backbone and side chain molar masses, an increase in the average number of grafted side chains significantly reduces the dependence of the damping function h(γ) on strain γ.

Graft-polystyrenes with side chains of similar molar masses Mw,br around 25 kg/mol, but varying average numbers of side chains per molecule are shown in Fig. 4b. A weaker damping with increasing average number of grafted side chains is found again. But comparing PS-80-0.6G-22 and PS-90-1.2G-27 with the damping function of PS-r-95, the influence of grafting is found to be not as strong as in the case of PS-60-1.4G-42 and PS-60-0.5G-55.

Two graft-polystyrenes with mass average molar masses of side chains well below the critical molar mass for entanglements Mc of linear chains and even well below the entanglement molar mass Me were also studied. Their shear damping functions are shown in Fig. 4c. The damping function of PS-105-6.7G-6 is very similar to that one for the linear PS-r-95. This result can be interpreted as such that the side branches which are very short compared to the molar mass for entanglements Me do not contribute to a changed strain dependence due to the lack of entanglements.

In contrast to this result, the graft-polystyrene PS-55-4.0G-13 with the longer side branches shows clearly a weaker dependence of h(γ) on strain than PS-105-6.7G-6. It can be compared to that one of the graft-polystyrenes PS-70-3.2G-22 or PS-60-1.4G-42. This result means that in addition to the length of the side chains another molecular property may influence the damping function.

Figure 5 shows the damping functions of all graft-polystyrenes together with the various number average mass fractions Φ n,br of grafted side chains. Three groups of graft-polystyrenes can be distinguished: the graft-polystyrenes with the highest mass fraction Φ n,br of about 0.50 which give the weakest damping and the graft-polystyrenes with a mass fraction Φ n,br between 0.14 and 0.30 which show a stronger damping. The third group is PS-105-6.7G-6 and PS-r-95 with the strongest damping. The similarity of h(γ) for PS-105-6.7G-6 with that of the linear PS-r-95 can be understood, as the short side chains of Mw,br=6,800 g/mol are well below the entanglement molecular weight of 18,000 g/mol and therefore do not affect the entanglement network. Somewhat surprising is the finding that h(γ) for PS-55-4.0G-13 and PS-70-3.2G-22 coincide with the other two samples of Φ n,br ≅ 0.5 although this have molar masses significantly below Mc. As a consequence it can be concluded that also short side branches with Me<Mw,br<Mc do alter the strain dependence of h(γ) and that the mass fraction Φ n,br of grafted side chains is the decisive parameter for the strain dependence of h(γ) for the grafted polystyrenes studied.

Uniaxial elongational flow

In elongational flow, the strain hardening and particularly its dependence on the molecular structure is of interest. Therefore, uniaxial elongational experiments were performed on the branched polystyrenes with varying molecular structure. The low-molecular weight residues of the branched PS samples could not totally be avoided by sample preparation (cf. Hepperle et al. 2005). To investigate their influence on the degree of strain hardening elongational experiments were performed in a way that for samples with different residue contents the temperature of the experiments was chosen to obtain a similar zero-shear viscosity. The results on two samples of different branching structure containing various amounts of residues are presented in Fig. 6. As it can clearly be seen, the strain hardening is independent of the content of low-molar mass residue.
Fig. 6

Influence of the low-molar mass residue content wMM on the uniaxial elongational viscosities. The measuring temperatures were adapted to match the zero shear viscosities of the materials: PS-60-2.1G-27: T=151.0° C, PS-55-2.1G-27: T= 147.5° C. PS-60-0.5G-55: T=158.0° C, PS-60-0.5G-55*: T=154.0° C. Lines denote 3η(t) from dynamic-mechanical data of graft-polystyrenes: (continuous line) 3η(t) with lower wMM, (dotted lines) 3η(t) with higher wMM. The lower curves are vertically shifted with the factor 0.3 for clarity

Influence of the number of grafted side chains

As it is well known from literature that besides branching high-molar mass components and the polydispersity can have an influence on strain hardening (Hepperle 2003; Münstedt and Laun 1981; Münstedt 1980) the graft-polystyrenes in this study were synthesized in a way that similar molar masses and molar mass distributions are obtained (Haug 2000).

The tensile stress growth coefficients \(\eta ^{+}_{\rm E} (t,\;\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon})\) of the melt of the linear PS-r-95 and those of the branched PS-90-1.2G-27 with a similar polydispersity are shown in Fig. 7. The zero-shear viscosities of the two melts are very similar and their strain hardening can therefore be compared directly. The linear PS-r-95 shows a slight strain hardening whereas the branched PS-90-1.2G-27 exhibits a pronounced one. The samples of PS-r-95 deform homogenously up to Hencky-strains of about 2.7. At higher strains the samples begin to neck-in slightly, which is indicated for the two lowest strain rates in Fig. 7, where at high strains the tensile stress growth coefficients are lower than the linear-viscoelastic viscosity 3η(t).
Fig. 7

Tensile stress growth coefficients \(\eta_{\rm E} ^{+}\;(t,\;\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon})\) of the linear PS-r-95 and the branched PS-90-1.2G-27 at measuring temperatures at which the zero shear viscosities are alike. Measuring temperatures: PS-r-95: T=169.0°C, PS-90-1.2G-27: T=168.0°C. The full lines denote 3η(t) from dynamic-mechanical data

Fig. 8

Tensile stress growth coefficients \(\eta_{\rm E} ^{+}(t,\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon})\) of three graft-polystyrenes with different average numbers of branches per molecule but similar grafted side chains. The full lines denote 3η(t) from dynamic-mechanical data

The influence of the average number of grafted side chains on strain hardening is investigated on a series of graft-polystyrenes with similar molar masses of the grafted side chains (Fig. 8a–c). The melt of the graft-polystyrene PS-70-3.2G-22 with an average of about three grafts per chain shows the highest strain hardening (Fig. 8a), followed by PS-60-2.1G-27 with an average of about two grafts per chain (Fig. 8b). The slightly branched PS-80-0.6G-22 (Fig. 8c) for which on average only every second molecule is branched, exhibits a lower degree of strain hardening than the other two graft-polystyrenes. From these results it can be concluded that a growing average number of grafted side chains leads to an increased amount of strain hardening, if melts of a similar topology are compared.

Influence of the molar mass of grafted side chains

Figure 9a, b display a comparison of the stress growth coefficients of melts of graft-polystyrenes with similar average numbers but different mass average molar masses of the grafted side chains. The measuring temperatures were adapted to obtain similar zero-shear viscosities for the different materials. Figure 9a shows the two weakly branched graft-polystyrenes with an average number of 0.5 and 0.6 branches per molecule. The graft-polystyrene PS-60-0.5G-55 has a similar mass average molar mass of the backbone (Mw,bb = 112,000 g/mol) compared to PS-80-0.6G-22 (Mw,bb=122,100 g/mol), but a mass average molar mass of the grafted chains which is a factor of 2.6 higher than for the graft-polystyrene PS-80-0.6G-22, resulting in a mass fraction Φ n,br of grafted chains that is twice as high. Despite the significantly higher molar mass of the grafted side chains, the melt of PS-60-0.5G-55 shows no significant increase in strain hardening compared to PS-80-0.6G-22. Only at the low strain rates of \(\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon} = 0.3\,s^{-1}\) and 0.1 s−1, the graft-polystyrene with the longer side chains exhibits a slightly higher strain hardening.
Fig. 9

Tensile stress growth coefficients \(\eta_{\rm E} ^{+}\;(t,\;\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon})\) of graft-polystyrenes with similar numbers of branches per molecule, but different mass average molar masses of grafted side chains. The full lines denote 3η(t) from dynamic-mechanical data. Temperatures were chosen to obtain similar zero shear viscosities for the different materials

Figure 9b displays the tensile stress growth coefficients for the melts of the graft-polystyrenes PS-60-1.4G-42 and PS-90-1.2G-27. The latter has a backbone mass average molar weight that is a factor of 1.5 higher (13.7×104 g/mol compared to 9.3×104 g/mol) and side branches that have mass average molar masses significantly lower, resulting in a mass fraction of grafted side chains of only 0.27 compared to a fraction of 0.49 for PS-60-1.4G-42. Despite the different topology, both melts show a similar degree of strain hardening. This result confirms that one obtained from the graft-polystyrenes with the lower amount of grafted side chains: not the length of the grafted side chains increases the amount of strain hardening, but the average number of grafts per molecule.

The tensile stress growth coefficients of the graft-polystyrenes with grafted side chains of a significantly lower molar mass than the critial molar mass Mc for entanglements are shown in Fig. 10. The side chains of PS-55-4.0G-13 have a mass-average molar mass Mw,br of 14,170 g/mol and the melt shows a pronounced strain hardening. The graft-polystyrene PS-105-6.7G-6 has even shorter side chains with a molar mass Mw,br of 6,800 g/mol. As the number of grafted side chains with about seven grafts per chain on average is higher, one might expect a higher degree of strain hardening, as the significance of the number of grafted chains for strain hardening has been demonstrated for other graft-polystyrenes (Fig. 8). However, the molar mass of the grafted side chains is much lower than the critical molar mass Mc for entanglements of linear chains and even well below the entanglement molecular weight Me. Due to the lack of entanglements the contribution to the elongational stress is therefore not as pronounced as for the graft-polystyrenes with entangled chains.
Fig. 10

Tensile stress growth coefficients \(\eta_{\rm E} ^{+}\;(t,\;\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon})\) of graft-polystyrenes with side chains of low molar mass. The full lines denote 3η(t) from dynamic-mechanical data

For all melts studied, the accuracy of the measurements is confirmed by the fact that the Trouton ratio ηE+(t) = 3 η(t) is fulfilled in the linear-viscoelastic range for all the strain rates applied. This is indicated in Figs. 7, 8, 9, and 10 by the solid lines representing 3η(t) derived from the dynamic data (Hepperle et al. (2005)) and shifted according to the WLF-equation with respect to the measuring temperature.

Quantification of the strain hardening properties

The strain hardening can be quantified using the strain hardening factor S defined as
$$S = \frac{{\eta ^{+}_{E} (t,\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon})}}{{3\eta (t)}}.$$

In comparison to the determination of S=f(ε) for one strain rate, the determination of \(\hbox{S}=\hbox{f}(\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon})\) at a fixed elongation allows one to account for the varying zero-shear viscosities of the different materials, as the strain hardening factors S can be compared at reduced strain rates \(\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon} \cdot \eta _{0} \). A similar suggestion was made by comparing strain hardening factors as a function of strain at a constant reduced strain rate \(\tau _{0} \cdot \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon}\) for different materials with τ0 being the terminal relaxation time (Takahashi et al. 1993). Values of S are determined from Figs. 7, 8, 9 and 10, for a strain of ε=3.0.

Figure 11a shows the strain hardening factors of the graft-polystyrenes with a similar mass average molar mass of the grafted side chains, but different average numbers \(\ifmmode\expandafter\bar\else\expandafter\=\fi{p}\) of grafted side chains per molecule. The strain hardening factor S increases with increasing average numbers \(\ifmmode\expandafter\bar\else\expandafter\=\fi{p}\) of grafted side chains per molecule. The lowest strain hardening is obtained for the melt of the linear polystyrene PS-r-95 which has a polydispersity comparable to the graft-polystyrenes (Hepperle et al. 2005). The two graft-polystyrenes PS-90-1.2G-27 and PS-80-0.6G-22 with a low average number of grafts per molecule show a higher strain hardening compared to the melt of the linear PS-r-95. PS-80-0.6G-22 and PS-90-1.2G-27 have comparable mass average molar masses of the backbone (Mw,bb=122,100 g/mol and Mw,bb=136,600 g/mol, respectively), but PS-90-1.2G-27 has twice the mass fraction Φ n,br of grafted side chains. The higher strain hardening of PS-90-1.2G-27 compared to PS-80-0.6G-22 can therefore be attributed to the higher average number \(\ifmmode\expandafter\bar\else\expandafter\=\fi{p}\) of grafted side chains per molecule. It is also evident from the stress growth coefficients shown in Fig. 11a that the slope of the dependence of the strain hardening factor S on the reduced strain rate \(\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon} \cdot \eta _{0} \) increases with the amount of grafted side chains.

A comparison of the higher branched graft-polystyrenes PS-70-3.2G-22 and PS-60-2.1G-27 reveals an obvious information about the role of the number of grafted chains per molecule on the strain hardening effect, as the number average mass fraction Φ n,br of grafted chains is constant. PS-70-3.2G-22 shows a more pronounced strain hardening than the graft-polystyrene with the lower number of grafted side chains (PS-60-2.1G-27) indicating once more the growing strain hardening with increasing number of grafted chains.

Figure 11b shows the comparison of the strain hardening factors for pairs of graft-polystyrenes with side chains of different molar masses. The graft-polystyrenes with a higher average number of grafted chains (PS-60-1.4G-42 and PS-90-1.2G-27) show slightly higher values for the strain hardening factor S than the graft-polystyrenes with lower average number of chains per molecule (PS-60-0.5G-55 and PS-80-0.6G-22).

Figure 11c presents the graft-polystyrenes with grafted side chains of molar masses below Mc and Me. The graft-polystyrene PS-105-6.7G-6 with short side chains (Mw,br=6,800 g/mol) does only show a slightly increased strain hardening factor compared to the linear PS-r-95, which can also be caused by the somewhat higher polydispersity of PS-105-6.7G-6 with Mw/Mn=1.9 in comparison to PS-r-95 with Mw/Mn=1.65. PS-55-4.0G-13 and PS-70-3.2G-22, however, possess a pronounced strain hardening. From these results follows that the side chains with Mw,br = 6,800 g/mol are too short to be entangled, whereas side chains with Mw,br=14,170 g/mol are already long enough to contribute to a higher stress level in uniaxial elongation.

Damping function in uniaxial elongational flow

With the assumption of separability of time and deformation, as experimentally found for many polymer melts (Wagner 1976; Wagner and Laun 1978), the stress tensor \(\underline{\underline \sigma} \) can be described by a single integral constitutive equation of the type
$$\underline{\underline \sigma} = - p \cdot \underline{\underline E} + {\int\limits_{- \infty}^t {m(t - {t}\ifmmode{'}\else$'$\fi) \cdot h({\text{I}}_{B}, {\text{II}}_{B}) \cdot \underline{\underline B} _{t} ({t}\ifmmode{'}\else$'$\fi)d{\kern 1pt}}}{t}\ifmmode{'}\else$'$\fi$$
p is the isotropic pressure, \(\underline{\underline E} \) the unit tensor and h(IB,IIB) the damping function with the invariants IB and IIB of the deformation tensor \(\underline{\underline B} _{\rm t}.\) The linear-viscoelastic memory function \(m(t - {t}\ifmmode{'}\else'\fi)\) is related to the viscoelastic relaxation modulus G(t) by
$$m(t - {t}\ifmmode{'}\else$'$\fi) = {\sum\limits_i {\frac{{g_{i}}}{{\lambda _{i}}}\exp {\left[ {- \frac{{(t - {t}\ifmmode{'}\else$'$\fi)}}{{\lambda _{i}}}} \right]}}} = \frac{{dG(t - {t}\ifmmode{'}\else$'$\fi)}}{{d{t}\ifmmode{'}\else$'$\fi}}.$$
with the discrete relaxation times λi and the relaxation strengths gi, as determined from the dynamic moduli (Hepperle 2003). With the inversion of Eq. 6 the damping function in uniaxial elongation can be calculated according to Wagner (1979) using
$$h_{u} (\varepsilon) = \frac{{\frac{{\sigma (\varepsilon)}}{{G(\varepsilon)}} - \frac{1}{{\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon}}}{\int\limits_0^\varepsilon {\sigma ({\varepsilon}\ifmmode{'}\else$'$\fi)\frac{{m({\varepsilon}\ifmmode{'}\else$'$\fi)}}{{G^{2} ({\varepsilon}\ifmmode{'}\else$'$\fi)}}d{\varepsilon}\ifmmode{'}\else$'$\fi}}}}{{\exp (2\varepsilon) - \exp (- \varepsilon)}}.$$

The calculated damping functions hu(ε) of the linear PS-r-95 and four branched graft-polymers with Φ n,br ≅ 0.5 are shown in Fig. 12. The damping functions hu(ε) are calculated for strain rates \(1.0 \geqslant \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon} \geqslant 0.1\,s^{-1}\) and deformations up to εH=3. Within experimental accuracy, the damping functions are the same for different strain rates for one material. A time-strain separability was already experimentally found for many branched and linear polymer melts (Wagner et al. 2000).

The linear PS-r-95 shows a damping function less dependent on strain than the predicition of the Doi–Edwards model (DE). This behavior is also reported by Wagner et al. (2000) for a PS melt with broader molar mass distribution (Mw/Mn=2.85). The damping functions of the branched polystyrenes shown in Fig. 12 are less dependent on strain than the melt of the linear PS-r-95. They are the less dependent on strain the higher the amount of strain hardening in unaxial elongation is (cf. Fig. 11). The damping functions h(γ) determined in shear flow (Fig. 5) are nearly the same for the four branched graft-polystyrenes shown in Fig. 12, but the damping functions hu(ε) in uniaxial elongation are significantly different. The damping functions determined via step-shear experiments can therefore not be used to predict the nonlinear-viscoelastic behavior in uniaxial extensional flow. The comparison of the damping functions h(γ) and hu(ε) also shows that uniaxial elongational flow is more sensitive to a varying branching structure than the stress responses determined with large deformations in shear flow.
Fig. 11

Strain hardening factor S as a function of reduced strain rate \(\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon}\eta _{0} .\) The dotted line for S=1 indicates the linear-viscoelastic behavior. aGraft-polystyrenes with varying average numbers of grafts per molecule. b Comparison of graft-polystyrenes with different molar masses of grafted side chains. cGraft-polystyrenes with grafted side chains of decreasing molar mass and increasing average number of grafts per molecule

Fig. 12

Damping functions of uniaxial flow hu(ε) calculated with Eq. 8 from uniaxial elongational viscosities and relaxation time spectra for linear PS-r-95 and graft-polystyrenes with Φ n,br ≅ 0.5. DE denotes the Doi–Edwards damping function in uniaxial elongation


The graft-polystyrenes studied here possess a certain polydispersity of the backbone. Therefore, all graft-polystyrenes also include topologies with a higher amount of branches than the average value, especially for molecules of higher molar masses resulting in a variety of structures like asymmetric stars, H-shaped structures and brushes. Their rheological properties show distinct differences in nonlinear-viscoelastic flows, depending on their molecular structure and on the type of flow applied to their melts. In uniaxial elongational flow, the tensile stress growth coefficient is mainly dependent on the average number of grafted chains per molecule, whereas the molar mass of grafted side chains does not significantly change the amount of strain hardening.

Surprisingly, rather short branches do show an influence on the damping function and on strain hardening. For example, the graft-polystyrene with Mw,br/Me = 0.8, i.e., PS-55-4.0G-13, still possesses an increased damping function in shear and in elongation and a higher degree of strain hardening than the linear PS with comparable polydispersity. However, if the side chains of the graft-polystyrene are very short (Mw,br/Me = 0.4 for PS-105-6.7G-6) the melt shows a damping function h(γ) similar to the one of a linear polystyrene, whereas in uniaxial deformation the strain hardening for this branched graft-polystyrene is only slightly higher than the one of a linear PS.

The damping function in elongation is less dependent on strain the higher the amount of strain hardening. The damping function in shear flow shows a dependence on the number average mass fraction of grafted side chains.

The comparison of the damping functions determined from step-strain experiments in shear flow and calculated with the relaxation time spectra and the extensional viscosities in uniaxial elongational flow shows that the influence of molecular topology on nonlinear-viscoelastic strain functions is quite different for various types of flow. Elongational flow is much more sensitive to differences in molecular structure than shear flow.



Financial support from the German Research Foundation (DFG) (grant numbers Mu 1336/2-1 and Mu 1336/2-3) is gratefully acknowledged. J.H. wants to thank Prof. Dr. M. H. Wagner for the source code of the program for the damping function calculations.

Supplementary material


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Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Institute of Polymer MaterialsUniversity Erlangen-NürnbergErlangenGermany
  2. 2.Bayer Technology Services GmbHLeverkusenGermany

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