Introduction

Isotactic polypropylene is one of the leading and fast growing thermoplastic polymers in the world due to its desirable and beneficial properties such as low density, high melting point, high chemical resistance, and low cost. However, commercial polypropylene produced by Ziegler–Natta or metallocene catalysts has linear chains and a relatively narrow molecular weight distribution [1–3]. Linear polypropylene has relatively low melt strength and exhibits no strain hardening, and as a result its application is restricted in processes where elongational properties, such as thermoforming, foaming, and blow molding are dominant [4, 5]. It is well known generally that the presence of branching improves the physical properties of polyolefins. Various investigations have shown that the phase behavior of polyethylene blends depends strongly on the presence of short or long branches. For the improvement of the melt properties of polypropylene, many attempts have been performed by some researchers [6–8]. Among these methods, introducing branches onto the backbone of commercial PP seems to be a more efficient approach because of its accessibility to industrialization and less cost [9]. Since the presence of LCB is generally known to enhance the melt strength of polymer, several approaches have been developed to synthesize long-chain branched polypropylene [10, 11]. Long-chain branches are commonly introduced to linear PP via electron beam or gamma irradiation [12, 13] and chemical modification [14, 15]. In the literature, the molecular structure and rheological properties of synthesized LCB-PPs has been investigated in details [16–18]. Compared to linear PP, LCB-PP shows a more pronounced shear-thinning behavior, a significant strain hardening and higher elasticity. The higher cost of LCB-PP leads researchers to use linear and long-chain branched polypropylene blends. In several cases, the foaming behavior of blends of linear and long-chain branched polypropylene were studied. Various studies have indicated that the foam morphologies of L-PP and LCB-PP blends are more uniform than those of neat L-PP. It was found that the strain hardening of branched polypropylenes caused high melt strength in addition to significant homogeneity of deformation in the elongational experiments [19, 20], which led to form foams with higher expansion ratios than L-PP [21].

In several cases the rheological behavior of blends of L-PP and LCB-PP has been investigated. It has been demonstrated that LCB-PP shows a more pronounced shear thinning; a higher elasticity and a significant strain hardening behavior compared to L-PP. Stange et al. [9] realized that the blending of L-PP with LCB-PP influences the rheological properties, especially the elongational behavior of L-PP. In addition, they found that the magnitude of strain hardening changed only slightly for the blends with higher than 25 wt% of LCB-PP, but a change of the dependency of the strain hardening on the strain rates occurred. McCallum et al. [22] concluded that the blending of L-PP and LCB-PP not only enhanced the melt strength, but the mechanical properties increased as well. Fang et al. [23] showed that increasing the length of short branches in metallocene LLDPEs with comparable molecular weights LDPE, could improve the miscibility of LLDPE/LDPE blends. Wagner et al. [24] investigated the elongational viscosities of blends of L-PP and LCB-PP and concluded that for the extruded LCB-PP a refining effect was observed. Tabatabaei et al. [25] found that the degree of crystallinity of the L-PP and LCB-PP blends first increased by adding small amounts of branched content but decreased by further addition.

This study aims to characterize the shear and elongational rheological properties of blends of L-PP and LCB-PP in detail as well as the applicability of “Molecular Stress Function” model for the blends. Our work focuses on the effect of molecular weight on the rheological properties of the blends. One of the novelties of this study is the verification of miscibility of polypropylene blends via rheological curves instead of other characterization methods due to its sensitivity to the molecular structure. The elongational viscosities of the blends were quantified with the MSF model.

Experimental

Materials

Four commercial linear polypropylenes (L-25, L-9, L-6, and L-3.2) were selected. The L-25 and L-6 were supplied by Iranian Shazand and Marun Petrochemical Companies with a melt flow rate (MFR) of 25 g/10 min (under ASTM conditions of 230 °C and 2.16 kg) and 6 g/10 min, respectively. The L-9 and L-3.2 were obtained from Iranian Jam Polypropylene Company and Regal Petrochemical Company with MFR of 9 g/10 min and 3.2 g/10 min, respectively. The commercial branched polypropylene (LCB-2.5) was produced by Basell. The molecular weights of the linear samples were obtained from the relation between the zero-shear viscosity and the molecular weight [26]. The main characteristics of testing materials are shown in Table 1. All PPs used in this work have a density about 0.9 g/cm3.

Table 1 Main characteristics of neat polypropylene grades
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A series of samples containing a range of LCB-PP contents (LCB-PP/L-PP weight ratios of 10/90, 25/75, 50/50, and 75/25) were prepared, as outlined below in the blend preparation section. All the pure samples were subjected to the same processing conditions in order to make sure that all of them have similar thermal history.

Blend preparation

Blends of L-PP and LCB-PP were prepared in a Brabender internal mixer at 210 °C and a rotor speed of 30 rpm for approximately 10 min (time to reach a constant torque). To avoid the thermal degradation of the polymers, 2000 ppm of an antioxidant (Irganox B225 from CibaGeigy) was used.

Rheological characterization

A hot press was used to form the compression molded disks that were 2 mm in thickness and 25 mm in diameter, at 200 °C. The dynamic melt rheological measurements were carried out using a controlled stress rheometer MCR-501 (by Anton Paar) in the oscillatory mode using parallel plate fixtures with 25 mm in diameter and a gap of 1.5 mm. All the measurements were carried out at 180 °C under a nitrogen atmosphere to limit degradation. Time sweeps confirmed that the samples did not degrade during the experiment. All the elongational viscosity measurements were performed using the SER HV-A01 model of the SERuniversal testing platform from Xpansion instrument. The rectangular samples with dimensions of 17 × 10 mm2 and thickness of about 0.5 mm were used.

Strain sweeps were performed to ensure that the measurements were within the linear viscoelastic regime. The storage modulus (\(G^{{\prime }}\)), loss modulus (\(G^{{\prime \prime }}\)) and complex viscosity (\(\eta^{ * }\)) were measured as a function of the angular frequency (\(\omega\)) at frequencies ranging from 0.02 to 600 rad/s.

Results and discussion

Rheological characterization of neat PPs

Figure 1 summarizes the complex viscosities of neat PPs as a function of angular frequency. The shear-thinning behavior of LCB-PP is more pronounced due to the presence of long-chain branches. In addition, with increasing the molecular weight of the L-PP the shear-thinning behavior is more pronounced and the transition zone from Newtonian plateau to the power-law flow region becomes smaller.

Fig. 1
figure 1

Complex viscosity as a function of frequency for neat PPs at T = 180 °C

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The plot of the loss angle, δ, as a function of frequency is plotted in Fig. 2. A uniform decrease in the loss angle is observed for the linear polypropylenes. At low frequencies the elasticity of the LCB-2.5 is larger compared to that of the L-PPs due to the presence of more entanglements as a consequence of long-chain branches. As a result of more shear-thinning behavior, the number of entanglements in the LCB-2.5 (Fig. 1) decreases with increasing frequency. At high frequencies the LCB-2.5 shows a flexion in the curve with a trend to a plateau. It has been shown that the extent and the breath of the plateau is related to the weight fraction of the branched chains [27].

Fig. 2
figure 2

Loss angle as a function of frequency for neat PPs at T = 180 °C

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The weighted relaxation spectra of the neat polypropylenes obtained using the NLREG (non-linear regularization) software [28] are shown in Fig. 3 to compare the relaxation behavior of the L-PPs and LCB-2.5 (the limit of frequencies within the experiment is shown as dashed lines). The area under the relaxation spectrum indicates the zero-shear viscosity of the melt which is reported in Table 1. To verify the accuracy of the spectra, the zero-shear viscosities of the samples obtained from the area under the relaxation spectra:

$$\eta_{0} = \int_{ - \infty }^{ + \infty } {\lambda H(\lambda ){\text{d}}(\ln \lambda )}$$
(1)

were compared with the Maxwell model

$$\eta_{0} = \sum {g_{i} \lambda_{i} },$$
(2)

where \(g_{i}\) and \(\lambda_{i}\) are the relaxation modulus and relaxation time, respectively. A good agreement between these data is observed (Table 1). As seen in Fig. 3, with increasing the molecular weight in the L-PPs the position of the peaks shifts to higher relaxation times. In addition, the LCB-2.5 displays a longer relaxation time than the L-PPs, illustrating that long-chain branches alter the stress relaxation mechanisms. Long-chain branches with slower motions such as arm retraction restrict the simple reptation that is expected for linear polymers.

Fig. 3
figure 3

Weighted relaxation spectra for neat PPs at T = 180 °C (dash lines represent the range of frequencies covered during experiments)

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Rheological characterization of the blends

The complex viscosities as a function of frequency for the L-25/LCB-2.5 and L-3.2/LCB-2.5 blends are plotted in Fig. 4a, b, respectively. These two blends are extreme cases and the other blends are intermediated. As seen in Fig. 4a, the complex viscosity of the L-25/LCB-2.5 blends at low frequencies is very small, a behavior which is the consequence of short molecular length of the L-25 that restricts the presence of entanglements. The complex viscosities of the blends at low frequencies increase dramatically with increasing the branched content. For the L-3.2/LCB-2.5 blends, with increasing the LCB-2.5 content shear-thinning is more pronounced.

Fig. 4
figure 4

Complex viscosity as a function of frequency for: a L-25/LCB-2.5, b L3.2/LCB-2.5 blends at T = 180 °C

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To investigate the miscibility of the blends, a logarithmic mixing rule has been applied to the blends. The logarithmic mixing rule is represented as [29]

$$\log \eta_{0} (\omega ) = \phi_{\beta } \log (\eta_{0} (\omega ))_{1} + (1 - \phi_{\beta } )\log (\eta_{0} (\omega ))_{2},$$
(3)

where \(\phi_{\beta }\) and \(\eta_{0}\) are the weight fraction of branched component and the zero-shear viscosity, respectively. For the L-25/LCB2.5 blends, as plotted in Fig. 5a, a good agreement with the logarithmic mixing rule is observed, suggesting that these blends are miscible. However, for the L-3.2 a deviation from the logarithmic mixing rule is observed for the branched contents higher than 50 wt% of the blends. This behavior has been reported for blends of linear and branched polypropylenes and polyethylenes in the literature [23]. It is believed that the molecular weights of the components as well as the amount of LCB content and the branching structure (e.g., tree-like or star-like) of the LCB component, influence the miscibility of the blends. Stange et al. [9] have demonstrated that the branching structure of the LCB-2.5 is tree-like and, as a consequence, represents a negative deviation from logarithmic mixing rule.

Fig. 5
figure 5

Zero-shear viscosity as a function of LCB-2.5 content at T = 180 °C (lines represent the mixing rule)

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For further investigation on the miscibility of the blends, the Cole–Cole plots (\(\eta^{{\prime \prime }} - \eta^{{\prime }}\)) of the blends are illustrated in Fig. 6. The radius of semi-circular shape is increased with increasing the molecular weight. For the L-25/LCB-25 blends the plots are intermediate, suggesting the miscibility of the blends. However, some synergistic effects for the L-3.2/LCB2.5 blends are obtained and no trend for the blends is observed. It is clear that with increasing molecular weight, the miscibility of the blends of L-PP and LCB-PP is reduced.

Fig. 6
figure 6

Cole–Cole plots for: a L-25/LCB-2.5, b L-3.2/LCB2.5 blends at T = 180 °C

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To investigate the effect of molecular weight of linear component on the relaxation behavior of the blends, the weighted relaxation spectra of the L-25/LCB2.5 and L-3.2/LCB-2.5 are shown in Fig. 7. It is clear that the addition of long-chain branched component alters the relaxation mechanism from the simple reptation, as expected for linear polymers, to the arm retraction, which postpones the movements of chains along their backbone; as a consequence the maxima in the curves shift to longer times and the spectrum shape becomes broader. As seen in Fig. 7a, for the L-25/LCB2.5, the peaks are intermediate to the neat polypropylenes, representing again miscibility; however, for the L-3.2/LCB-2.5 blends no specific trend is observed.

Fig. 7
figure 7

Weighted relaxation spectra for: a L-25/LCB-2.5, b L-3.2/LCB2.5 blends at T = 180 °C

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Only a few models have been developed to predict the linear viscoelastic behavior of polymer blends to correlate the viscoelastic properties of emulsion systems to microstructure. Furthermore, these models can correlate the dynamic response of polymer blends to the interfacial tension between their components and the morphology composition.

A model for predicting the linear viscoelastic properties of some immiscible emulsion-type blends has been developed by Palierne [30]. For a narrow distribution of incompressible droplet diameters as well as constant interfacial tension, the complex modulus of a blend is expressed by

$$G_{\text{B}}^{*} (\omega ) = G_{\text{m}}^{*} (\omega )\frac{{1 + 3\phi H^{*} (\omega )}}{{1 - 2\phi H^{*} (\omega )}}$$
(4)

\(H^{*}\) is defined as

$$H^{*} = \frac{{\left( {\frac{\alpha }{{R_{v} }}} \right)(2G_{\text{m}}^{*} (\omega ) + 5G_{\text{d}}^{*} (\omega )) + (G_{\text{d}}^{*} (\omega ) - G_{\text{m}}^{*} (\omega ))(16G_{\text{m}}^{*} (\omega ) + 19G_{\text{d}}^{*} (\omega ))}}{{10\left( {\frac{\alpha }{{R_{v} }}} \right)(G_{\text{d}}^{*} (\omega ) + G_{\text{m}}^{*} (\omega )) + (3G_{\text{m}}^{*} (\omega ) + 2G_{\text{d}}^{*} (\omega ))(16G_{\text{m}}^{*} (\omega ) + 19G_{\text{d}}^{*} (\omega ))}}$$
(5)

\(\alpha\) is the interfacial tension, \(\phi\) is the volume fraction of the droplets with volume average radius of \(R_{v}\), \(G_{\text{d}}^{*}\) and \(G_{\text{m}}^{*}\) are the complex moduli of the droplets and matrix, respectively. The storage and loss moduli of blends can be easily calculated using some simple algebraic equations [31]. These models, which have been modified for melt blends, show a good agreement with experimental data [32].

Measuring the dimension of droplets in the L-PPs and LCB-PP blends is impossible due to the low optical contrast between the PP components. Tabatabaei et al. [25] suggested \(\alpha /R_{v}\) as a single parameter to find the best fits of experimental data for the blends. Here the influence of \(\alpha /R_{v}\) (0 and 500) on the storage modulus for the blends of the L-25 and L-3.2 with 75 wt% LCB-2.5 is predicted by the Palierne model which the results are shown in Fig. 8a, b, respectively. For the blends of the other linear components (L-6 and L-9) with 75 wt% of LCB-2.5, the same trend as L-25 is obtained. In comparison, the predicted \(G^{{\prime }}\) values at low frequencies are influenced by the values of \(\alpha /R_{v}\). As shown in Fig. 8a, for the (25/75) L-25/LCB2.5 blend the value of \(\alpha /R_{v}\) = 0 achieves a better fit of the model to the experimental data; however, for the (75/25) L-3.2/LCB2.5 blend, the value of \(\alpha /R_{v}\) = 500 is closer to the experimental data. The zero value of interfacial tension is an indication of miscibility, while other rheological investigation suggests the miscibility of the L-25/LCB2.5 blends. However, the non-zero value of interfacial tension seems to be a demonstration of the presence of a dispersed phase.

Fig. 8
figure 8

Comparison between experimental data and prediction of storage modulus using Palierne model for: a (25/75) L-25/LCB2.5 blend, b (25/75) L-3.2/LCB2.5 blend at T = 180 °C

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It should be noted that in immiscible blends the storage modulus at low frequencies increases while the loss modulus does not differ significantly [25]. The values of \(G^{{\prime \prime }}\) predicted by the model are not sensitive to the values of \(\alpha /R_{v}\) and are in good agreement with the experimental data.

Extensional data are very sensitive to the molecular structure. Several viscoelastic models can describe the growth of stress during the uniaxial elongation of polymeric melts. By generalizing the Doi–Edwards model to the Molecular Stress Function (MSF) a successful prediction of the rheological behavior of linear and long-chain branched polymer melts in transient uniaxial constant extension-rate experiments is possible [33–35]. In the MSF model, the relative tension of the macromolecular chain in a tube segment of diameter, \(a\), at time, \(t\), is expressed by a Molecular Stress Function:

$$f(t,t^{\prime} ) = a_{0} /a(t,t^{\prime} )$$
(6)

\(t^{{\prime }}\) represents the time when the tube was created with equilibrium tube diameter \(a_{0}\). The extra stress tensor is given by

$$\sigma (t) = \int_{ - \infty }^{t} {m(t - t^{\prime} )f^{2} } (t,t^{\prime} )S_{\text{DE}}^{\text{IA}} (t,t^{\prime} ){\text{d}}t^{\prime},$$
(7)

where

$$m(t - t^{\prime} ) = \frac{{\partial G(t - t^{\prime} )}}{{\partial t^{\prime} }} = \sum\limits_{i = 1}^{N} {\frac{{g_{i} }}{{\lambda_{i} }}} e^{{ - \frac{t - t'}{{\lambda_{i} }}}}$$
(8)

is the Maxwell memory function. According to the MSF model, the tension in a chain segment is related to the deformation history of the tube segments. \(S_{\text{DE}}^{\text{IA}}\) indicates the strain measure of the Doi-Edwards model, using the independent alignment (IA) assumption:

$$S_{\text{DE}}^{\text{IA}} = 5f^{2} \left\langle {\frac{{{\text{u}}^{{\prime }} {\text{u}}^{{\prime }} }}{{u^{{{\prime }2}} }}} \right\rangle = 5S_{\text{DE}},$$
(9)

where \(S_{\text{DE}}\) is the second-order orientation tensor and \(\left\langle \cdots \right\rangle\) denotes an average over an isotropic distribution function of unit vectors, and \(u^{\prime}\) is the length of the unit vector \({\mathbf{u}}^{ '}\).

In elongational deformations the evolution equation is expressed by

$$\frac{{\partial f^{2} }}{\partial t} = \mathop \varepsilon \limits^{ \cdot } \frac{{\beta f^{2} }}{{1 + \frac{\beta - 1}{{f^{4} }}}}\left( {S_{11} - S_{33} - \frac{{f(f^{3} - 1)}}{{f_{\hbox{max} } (f_{\hbox{max} }^{3} - 1)}}\sqrt {S_{11} + \frac{1}{2}S_{33} } } \right)z$$
(10)

\(S_{11}\) and \(S_{33}\) are the respective components of the orientation tensor parallel and perpendicular to the direction of elongation. It should be noted that the parameter \(\beta\) describes the topology of the melt and indicates the slope of the transient elongational viscosity curve. When the tube is stretched, one segment is extended, while \((\beta - 1)\) segments are compressed.

The time-dependent elongational viscosities (\(\eta_{E}^{ + } (t,\mathop {\varepsilon_{0} }\limits^{ \cdot } )\)) of the L3.2/LCB2.5 blends are presented in Fig. 9a–f and compared to the prediction of the MSF model. For the linear polypropylene, all the curves measured at different strain rates have superimposed on the linear viscoelastic curve. The transient elongational viscosity is calculated by

$$\eta_{\text{E}}^{ + } = 3\eta^{ + } = 3\sum\limits_{i} {\frac{{g_{i} }}{{\lambda_{i} }}\left[ {1 - \exp \left( { - \frac{t}{{\lambda_{i} }}} \right)} \right]},$$
(11)

where \(g_{i}\) and \(\lambda_{i}\) are the discrete relaxation spectra calculated from the Maxwell model and are shown for the L-3.2 blends in Table 2.

Fig. 9
figure 9

Stress growth coefficients data in extension and prediction of MSF model for: a L-3.2, b 10 wt% LCB-2.5, c 25 wt% LCB-2.5, d 50 wt% LCB-2.5, e 75 wt% LCB-2.5, f LCB-2.5 at 180 °C

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Table 2 Relaxation spectra of L-3.2 blends at T = 180 °C
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A significant strain hardening takes place upon addition of low amounts of the LCB-PP. Deviations from the linear viscoelasticity at shorter times and sharper increase in the elongational viscosity were found as the weight fraction of the LCB-PP increase.

A good agreement between the elongational viscosities and MSF model is observed. The range of values of the \(\beta\) parameter in this work varies from 1 for the L-3.2 to 2.2 for the LCB-2.5 which are in accordance with 1 for the linear polymer and 2 for the long-chain branched polymer, respectively [36]. Furthermore, \(f_{\hbox{max} }^{2}\) which represents the maximum amount of energy stored in a tube segment, increases with amount of LCB-2.5 in the blends.

Conclusions

In this study, the shear and extensional rheological properties of the blends of L-PPs having different molecular weights and LCB-PP were investigated. Based on the log-additivity rule, for the zero-shear viscosity, weighted relaxation spectra, Cole–Cole plots and Palierne model, the blends of low molecular weight linear polypropylene showed miscibility while for high molecular weight L-PPs and higher amount of LCB-PP, immiscibility was observed. The Palierne model predictions were in good agreement with the experimental data.

With adding a small amount of LCB-PP, significant strain hardening was observed. The Molecular Stress Function could predict the strain hardening behavior of the blends.