Slip at the interface between immiscible polymer melts I: method to measure slip

Typically, two chemically different polymers are immiscible, and the distinct phases are separated by interfaces. At the interface between two phases, the chemically different chains are likely to be weakly entangled. If entanglements are the main source of adhesion between the two components, then poor interfacial adhesion can be expected at the interface. During coextrusion of two immiscible polymeric liquids, as the flow rate is increased, the stress at the interface progressively increases. Due to the poor interfacial adhesion mentioned earlier, the interface would eventually be unable to support the interfacial stress. Under such conditions, one of the phases might slip with respect to the other. Such slip would also imply that the usually assumed no-slip boundary condition (Goldstein 1938) is no longer valid at such a liquid/liquid interface.

In spite of the rather simple scenario described above, the possibility of slip at the interface between two polymers remained rather controversial for a long time. At least part of the reason for the reluctance could be attributed to the success of the no-slip boundary condition in describing multiphase flows of Newtonian liquids (Goldstein 1938). However, during pressure-driven multilayer flow of polystyrene/polyethylene (both high density and low density), Han and Chin (1979) found that the pressure gradient at a given flow rate was even smaller than that of either of the two component polymers extruded individually. The authors hypothesized slip at the polymer/polymer interface in order to rationalize these puzzling results. Later, Bousmina et al. developed a model for the effective viscosity of immiscible blends of (generalized) Newtonian fluids that included the effect of slip at the liquid/liquid interface (Bousmina et al. 1999).

Among experimental techniques to study polymer/polymer interfacial slip, optical techniques have proved rather powerful (Migler et al. 2001; Lam et al. 2003; Zartman and Wang 2011). In such experiments, the slip velocity at the polymer/polymer interface can be obtained from the difference between the measured velocities of the incorporated particles that can be observed on either side of the polymer/polymer interface (Migler et al. 2001; Lam et al. 2003; Zartman and Wang 2011). Optical techniques are conceptually simple and yield accurate measurements of the slip velocity with few assumptions (Migler et al. 2001; Lam et al. 2003; Zartman and Wang 2011). However, the technique involves observing particles suspended in the flowing liquid and this requires equipment made of materials transparent to visible light. Given this, methods to determine the slip velocity at a liquid/liquid interface using only rheological measurements could prove useful.

Over the past decade, several groups have used rheological techniques to infer the slip velocity at a polymer/polymer interface (Zhao and Macosko 2002; Jiang et al. 2003; Lee et al. 2009; Park et al. 2010; Zhang et al. 2006). Most of these studies are restricted to multilayers with the parallel layers stacked on top of each other. For instance, Macosko and coworkers (Zhao and Macosko 2002; Lee et al. 2009; Park et al. 2010; Zhang et al. 2006) have estimated interfacial slip velocities using the reduction in the measured viscosity in the presence of slip when compared to the value estimated in the absence of slip. The viscosity in the absence of slip was determined as the volume fraction weighted harmonic mean of the viscosities of the individual polymers. While this estimate for the no-slip viscosity is appropriate for a stack of parallel multilayers, devising such estimates for other relevant geometries (such as coaxial flow during coextrusion) is not straightforward. The mixing rules (Utracki 1989; Bousmina et al. 1999) that are used for such estimations are usually at least partly empirical and their range of validity and accuracy are not always clear. Hence, methods for the determination of the slip velocity at a polymer/polymer interface directly from rheological data are few, especially for cases where reliable estimation of the no-slip viscosity is not straightforward. This lack is particularly troublesome in the case of coextrusion because strain rates significantly higher than that during plane Couette steady shear flow (Lee et al. 2009; Park et al. 2010) are routinely accessed.

In this first of a series of planned articles, we propose a novel method to estimate the slip velocity at a polymer/polymer interface during two-phase coaxial flow. The method is a modification of the Mooney method and allows an approximate but sufficiently accurate determination of the slip velocity at the interface. To determine the validity of our method, we compare the slip velocity determined using the modified Mooney method to that obtained using the reduction in the viscosity due to slip when compared to the value obtained assuming no-slip. To render the estimation of the viscosity under the assumption of no-slip straightforward, we choose isotactic polypropylene (PP) and atactic polystyrene (PS) with almost identical viscoelastic properties over a range of shear rates (see Fig. 1). We then provide a comparative discussion of the results obtained from the two methods and finally, conclude.

First, we will describe the deviation from no-slip method to obtain the slip velocity (method (a)). Figure 3 shows the flow curves for the two pure melts (PP and PS) and the core–sheath PP/PS samples along with the Carreau-Yasuda fit for the pure melts. In this study, we measured V _s−i using capillary dies of three different diameters D = 1. 0, 1.5 and 2.0 mm but with fixed L/D = 40 and an entry angle of 60°. Therefore, the shear stress and shear strain rate shown in Fig. 3 are apparent values (i.e, the Bagley correction and the Weissenberg-Rabinowitsch correction were not performed to determine the V _s−i). As the stress and strain rate are apparent values, the λ and a of the Carreau-Yasuda fit in Fig. 3 are slightly different from the parameters in Fig. 1b. The parameters for the Carreau-Yasuda fit shown in Fig. 3 are η ₀ = 8595 Pa s, n = 0. 33, λ = 0. 41, and a = 0. 71. In order to evaluate the V _s−i using the deviation from the no-slip condition (4), we used the parameters for the Carreau-Yasuda fit shown in Fig. 3. As Fig. 3 indicates, at low interfacial stress values, the apparent shear rates of the two-layer PP/PS samples display good agreement with that of the individual melts and the Carreau-Yasuda fit. However, above a certain critical shear stress, the apparent shear rate at a given value of shear stress deviates from the Carreau-Yasuda values indicating the existence of interfacial slip. Now, Eq. 4 can be used to determine the V _s−i as a function of σ _i and the results are shown in Fig. 5.

To obtain V _s−i using method (b), the quantity Q _PP/PS/πR ³ is plotted against 1/R (which is equivalent to 1/R _i) for various (constant) values of σ _w (or equivalently, σ _i or the pressure drop). Such plots are shown in Fig. 4a for PP, Fig. 4b for PS, and Fig. 4c for the core–sheath PP/PS samples. From Fig. 4a, b, it is clear that the slopes of the Q _PP/πR ³ and Q _PS/πR ³ versus 1/R lines are both approximately zero. These results imply that for σ _w < 10⁵ Pa there is negligible slip at the wall of the capillary die for both the PP and the PS melts. On the other hand, the slope of the line in Fig. 4c increases above a certain σ _i. This suggests the existence of slip in the core–sheath PP/PS samples. As the slip at the wall is negligible for both the PP and the PS melts, the likely origin of the observed slip is at the interface between the PP and the PS in the core–sheath samples. This is also the reason for attributing the deviation in the shear rate from the Carreau-Yasuda values to slip at the polymer/polymer interface, as was done earlier. The data in Fig. 4c in combination with Eq. 7, which together constitute the modified Mooney method, can be used to determine V _s−i from the slope of the Q _PP/PS/πRi\(\left (\text {and} Q_{\text {PP}/\text {PS}}/\pi R_{\mathrm {i}}^{3}\right )\) ³ versus 1/R _i line. Using (R − R _i )/R ≈ 2. 3 × 10⁻² and propogating the error leads to a maximum relative error in the determination of the V _s−i of approximately 4.5 × 10⁻². Consequently, the error bars in Fig. 5 are all smaller than the symbol size. This suggests that the error in the determination of the interfacial slip velocity introduced by the approximations inherent in the modified Mooney method are likely to be negligible and hence can be ignored.

While investigating wall slip, Hatzikiriakos et al. (1993) examined the effect of coating fluropolymers (FP) on the walls of a sliding plate rheometer on the observed slip of linear low density polyethylene (LLDPE) melts. The experimental procedure used by Hatzikiriakos et al. is conceptually similar to that used in our work but the flow geometry to which the method is applied is different. More importantly, in contrast to Hatzikiriakos et al., we exclusively focus on the slip that occurs at the interface between two immiscible polymers in the melt state. Further, in the experiments of Hatzikiriakos et al., the thickness of the FP layer, h _i, was approximately 2 μm and the gaps between the sliding plates, H, was either 0.36 or 0.23 mm. This approximation contributes to a relative error in the gap width, ( h _i(top) + h _i(bottom))/H ≈ 1 − 2 × 10⁻². Interestingly, this relative error is comparable to the relative error in our work, ((R − R _i)/R ≈ 2. 3 × 10⁻²).

Figure 5 shows a comparison between the V _s−i determined using Eq. 4 and that from Eq. 7. The figure suggests that, at a given σ _i, V _s−i determined from both Eqs. 4 and 7 are rather close. This suggests that Eq. 7 is an appropriate method to estimate the slip velocity at a polymer/polymer interface. The advantage of the modified Mooney method is that it can used to determine the interfacial slip velocity in core–sheath systems even when polymers with different shear rate-dependent viscosities are coextruded. Further, the slip at the PP/PS interface occurs at shear stress values that are significantly lower than the shear stress necessary for the onset of wall slip ( ≈ 10⁵ Pa) (Hatzikiriakos and Dealy 1992; Mitsoulis et al. 2005; Komuro et al. 2010). If both wall slip and interfacial slip interfacial slip are present, it is not possible to determine the individual contributions to the total slip velocity using only the modified Mooney method. Hence, for the two polymers investigated here, the polymer/polymer interfacial slip velocity for σ _w > 10⁵ Pa cannot be obtained.

It is also germane to note that the lower and upper limits of the range of σ _i over which the V _s−i can be determined depend on both the diameter of capillary die and the pressure gradient. While using the deviation from no-slip method, i.e., Eq. 4, measurements from a single capillary die are sufficient to determine V _s−i. As we used three capillary dies with D = 1. 0, 1. 5, and 2.0 mm, the range of σ _i over which the V _s−i can be obtained is given by the union of the range of σ _i accessible by the individual dies and hence is relatively wide (see Fig. 5). On the other hand, while using the modified Mooney method, i.e., Eq. 7, V _s−i for a given σ _i can be evaluated only when that σ _i is accessible by all three of the dies. Hence, the range of σ _i over which the V _s−i can be measured using the Mooney method is given by the intersection of the range of σ _i that can be measured by each of the three dies and hence is relatively narrower (see Fig. 5).

Figure 5 indicates that the dependence of V _s−i on σ _i is a power-law. Depending on the value of the interfacial stress, the power-law exhibits two distinct regimes with the power-law exponent changing from approximately 3 (region A) to approximately 2 (region B) at a critical interfacial stress of approximately 2 × 10 ⁴ Pa. The low-stress power-law exponent of 3 is in reasonable agreement with that found by Lee et al. (2009) for slip at the PP/PS interface. The high-stress exponent of 2 is similar to that found for a different polymer pair by Migler et al. using stroboscopic optical microscopy (Migler et al. 2001) and Park et al. using rheological measurements (Park et al. 2010). However, the actual V _s−i values for PP/PS obtained in this study are consistently (approximately two to three times) larger than previous results (Zhao and Macosko 2002; Lee et al. 2009). A possible reason for the discrepancy is discussed below.

Recently, we also investigated polymer/polymer interfacial slip during coextrusion of a sandwich three-layer sample using a slit die (Komuro et al. 2013). These experiments were performed using the same set of samples used in the current study and at the same temperature (230 °C). As discussed there, the power-law relationship between the interfacial slip velocity and the interfacial stress is essentially unchanged if the polymer in the inner layer is switched with that in the outer layer (Komuro et al. 2013). In addition, it was found that the interfacial slip velocity measured in the slit die using the PP/PS/PP and the PS/PP/PS layer stackings were rather close to the interfacial slip velocity measured in the capillary die using the concentric core–sheath PP/PS sample. These results suggest that the power-law relationship between V _s−i and σ _i and the actual value of V _s−i at a given σ _i are essentially independent of the flow geometry. In light of the independence of the value of the interfacial slip velocity on the flow geometry, one possible reason for the discrepancy in the V _s−i values for PP/PS obtained in this study under two-phase coaxial flow and previous results in the literature on sandwich multilayer samples (Zhao and Macosko 2002; Lee et al. 2009) is the temperature at which the experiments were performed. As Zhao et al. and Lee et al. performed their experiments at 200 °C, it is reasonable that our experiments at 230 °C yield a higher interfacial slip velocity at a given interfacial stress. Preliminary measurements at 200 °C indicate that the interfacial slip velocity obtained from the modified Mooney method agree well with the values of Macosko and coworkers (Zhao and Macosko 2002; Lee et al. 2009). The effect of temperature on polymer/polymer interfacial slip will be further discussed in the planned third part of this series of articles.

Extrapolating the data of Lee et al. (2009) for PP/PS to where it intersects with the data of Migler et al. (2001) for polyethylene/fluoropolymer (PE/FP), the interfacial shear stress at the transition between the two power-law regimes can be estimated to be approximately 6 × 10⁴ Pa. Lee et al. (2009) also present data for PE/FP pair and using the high shear stress data of Migler et al. (2001) suggest that the transition occurs at 5 × 10⁴ Pa. In addition, independent measurements for a PE/FP interface by Park et al. (2010) found that the stress at the transition was approximately 5 × 10⁴ Pa. Although these experiments were performed using different polymer pairs, the value at the transition compares favorably with that determined from Fig. 5. When combined with our results, these results suggest that the value of the interfacial stress at the power-law transition is not sensitively dependent on the chemistry of the particular polymer pair being investigated. These results are certainly intriguing and we hope to shed additional light into this aspect in a future publication.

As pointed out in previous studies (Migler et al. 2001; Lee et al. 2009; Park et al. 2010), the values of the power-law exponent in the two regions disagree with existing theories including the classical predictions of (Brochard-Wyart and de Gennes 1993). In addition, the transition between the power-law regimes is a smooth crossover instead of the sudden jump envisaged in their predictions. The origin of the discrepancy is currently unclear. One reason suggested for the discrepancy between the theory and experiment is that all of the experiments, including ours, have used polymers with significant polydispersities. We hope that future studies using relatively monodisperse polymer pairs, if feasible, can shed light on this issue. Even without such direct corroboration of the theory, it is possible that the change in the power-law exponent between the two regimes corresponds to a change in the mechanism of slip at the polymer/polymer interface, similarly to that envisioned by Brochard-Wyart and de Gennes (1993). Investigations regarding the mechanisms of interfacial slip and also the effect of varying the viscosity of the two polymer components are also currently in progress.

We have investigated slip at a polymer/polymer interface during two-phase coaxial flow of PP and PS using two methods. To enable the determination of the slip velocity by measuring the deviation from no-slip, we choose two polymers with similar viscoelastic properties over a wide range of shear rates. On the one hand, such a choice drastically simplifies the estimation of viscosity of the concentric core–sheath sample under no-slip. On the other hand, choosing two polymers with matched viscosities allows us to avoid considering possible effects of the viscosity difference on polymer/polymer interfacial slip. In addition, we have introduced a method to determine interfacial slip velocity in two-phase coaxial flow. The method is a modification of the Mooney method traditionally used to study wall slip. One advantage of this method is that, in principle, it is straightforward to apply even when the viscosities of the two polymers being coextruded are different. To establish the validity of the method, we compared the interfacial slip velocity determined by this method to that determined by the more conventional deviation from no-slip method. The slip velocity at the polymer/polymer interface V _s−i, as determined by the modified Mooney method agreed rather well with that determined by the deviation from no-slip. Further, slip at the polymer/polymer interface occurs at values of shear stress significantly lower than that necessary for the onset of wall slip ( ≈ 10⁵ Pa).

The slip velocity at a polymer/polymer interface exhibits a power-law dependence on the interfacial shear stress. By measuring the interfacial slip velocity over a wide range of shear stress values, we were able to directly observe that the power-law exponent changes from approximately 3 to approximately 2 at a critical stress of approximately 2 × 10 ⁴ Pa. Combining the results of this study with that of Komuro et al. (2013), there exists strong evidence that neither the power-law relationship between V _s−i and σ _i and nor the actual value of V _s−i at a given σ _i strongly depend on the flow geometry.

Our results suggest that the modified Mooney method could be profitably used to estimate the slip velocity at a polymer/polymer interface. A significant advantage of the proposed method is that it can be used to investigate interfacial slip under two-phase coaxial flow where the two polymers possess different viscosities. Such an estimation is possible because the method does not rely on determining the volumetric flow rate under no-slip and consequently, mixing rules for the viscosity are not necessary.