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Laura
Casanellas‡
^{ab},
Manuel A.
Alves
^{c},
Robert J.
Poole
^{d},
Sandra
Lerouge
^{b} and
Anke
Lindner
*^{a}
^{a}PMMH, UMR 7636 CNRS – ESPCI Paris – Université Pierre et Marie Curie Université Paris Diderot, 10 rue Vauquelin, F-75231 Paris CEDEX 05, France. E-mail: anke.lindner@espci.fr
^{b}Laboratoire Matière et Systèmes Complexes, CNRS UMR 7057-Université Paris Diderot, 10 rue Alice Domond et Léonie Duquet, 75205 Paris CEDEX 05, France
^{c}Departamento de Engenharia Química, CEFT, Faculdade de Engenharia da Universidade do Porto, Rua Dr Roberto Frias, 4200-465 Porto, Portugal
^{d}School of Engineering, University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK

Received
8th February 2016
, Accepted 16th May 2016

First published on 17th May 2016

We determine both experimentally and numerically the onset of elastic flow instabilities in viscoelastic polymer solutions with different levels of shear thinning. Previous experiments realized in microfluidic serpentine channels using dilute polymeric solutions showed that the onset of elastic instabilities strongly depends on the channel curvature. The scaling dependence is well captured by the general instability scaling criterion proposed by Pakdel and McKinley [Phys. Rev. Lett., 1996, 76, 2459:1–4]. We determine here the influence of fluid shear thinning on the onset of such purely-elastic flow instabilities. By testing a set of polyethylene oxide solutions of high molecular weight at different polymer concentrations in microfluidic serpentine channels we observe that shear thinning has a stabilizing effect on the microfluidic flow. Three-dimensional numerical simulations performed using the White–Metzner model predict similar trends, which are not captured by a simple scaling analysis using the Pakdel–McKinley criterion.

Most previous studies on purely-elastic flow instabilities, in a sensible attempt to separate out effects of elasticity from shear thinning, have used constant-viscosity elastic liquids, so-called Boger fluids.^{4–6} However, the vast majority of viscoelastic fluids, both in nature (e.g. blood,^{7} synovial fluid or hyaluronic acid,^{8} snail mucus^{9}) and synthetic fluids used in different industries (polymer solutions^{10} or wormlike surfactant solutions used in personal products,^{11} foodstuffs or pharmaceutical and chemical settings^{12}), often display strong shear thinning of the shear viscosity, especially at high shear rates. Physically, shear thinning arises as a direct consequence of the fluid microstructure, especially when the fluid can no longer be considered to be dilute (in some sense) and interactions occur: in colloidal suspensions it may arise via alignment of suspended particles under flow; in polymeric fluids or wormlike micellar solutions via alignment or disentanglement.

Previous studies which have used dilute polymer solutions, i.e. essentially Boger fluids, have revealed much about onset conditions, physical mechanisms and scaling of elastic flow instabilities. In particular, it is now commonly accepted that the main mechanism responsible for the appearance of most purely-elastic instabilities is a combination of flow-induced tensile stresses and the streamline curvature (elegantly captured in the Pakdel–McKinley criterion^{13}). For example, Zilz et al.^{14} were able to use the Pakdel–McKinley criterion to systematically test the effect of the flow curvature on the appearance of purely-elastic instabilities in microfluidic serpentine (wavy) channels for dilute polymer solutions. Good agreement was observed between the analysis based on simple scaling arguments, microfluidic experiments and three-dimensional numerical calculations using the upper-convected Maxwell (UCM) model. The aim of the current paper is to extend this analysis to incorporate more realistic rheology, specifically shear-thinning effects.

To date, the influence of non-linear rheology, and in particular shear thinning of the shear viscosity, on the onset of purely-elastic flow instabilities has not been studied in detail. McKinley et al.^{15} discussed how shear thinning, solvent viscosity and spectra of relaxation times might be incorporated into the instability criterion considering cone-and-plate flows (i.e. where the shear rate and the radius of curvature are known). We will show here that the correction suggested to incorporate shear-thinning effects only works for small amounts of shear thinning and fails at higher levels. Other studies which have probed shear-thinning effects include the works of Larson et al.^{16} who undertook a combined linear stability analysis with the K-BKZ model and experimental measurements in the Taylor–Couette geometry. They found shear thinning tended to increase the critical Deborah number, defined as the product between the shear rate and the longest relaxation time, while decreasing the critical value of the ratio between the first normal-stress difference (N_{1}) and the shear stress. The experiments exhibited instabilities that occur at shear rates 20–45% lower than those predicted by the axisymmetric linear stability analysis but this discrepancy was attributed to the appearance of non-axisymmetric modes. The effect of polymer concentration, which incorporates changes in shear thinning once out of the dilute regime, on the onset of elastic turbulence in Taylor–Couette flow was studied by Jun and Steinberg.^{17} More recently, Nicolas and Fuchs^{18} predicted, using the White–Metzner model, a stabilizing effect on the onset of elastic flow instabilities in dense colloidal suspensions, which was attributed to the significant shear-thinning behaviour of such systems. Theoretical investigations performed by Wilson and Rallison^{19} using also the White–Metzner model showed that for strongly-shear-thinning fluids (with a power-law exponent, n < 0.3) the flow could become elastically unstable in a straight channel (in the absence of a curvature), due to the large degree of shear thinning of the fluid. These predictions were supported by experimental work done by Bodiguel et al.^{20} using polymer solutions.

As touched on briefly above, in this work we aim to assess the influence of shear thinning (0.6 < n < 1) on the onset of a purely-elastic flow instability in microfluidic serpentine channels. The use of microfluidic channels enables us to use water-based fluids, which are inherently cheap, safe and easily disposed of, whilst still keeping inertial effects small. In particular, polymeric solutions are of great interest since they have been extensively characterized in the literature.^{21–30} We are able to systematically modify the degree of shear thinning by tuning the polymer concentration and, in particular, by crossing the critical overlap concentration into the semi-dilute regime. As the foregoing has made clear, our results have many important potential applications – especially for ‘real’ fluids which are invariably shear thinning – as well as identifying a possible limitation of the Pakdel–McKinley scaling criterion which suggests the need for a generalization of the theoretical framework to predict the experimental findings observed here.

Rheological measurements presented in Section 3 were performed using both a strain-controlled rheometer (ARES-G2, from TA Instruments), with a 50 mm diameter and 2.3° titanium cone geometry, as well as a stress-controlled rheometer (Physica MCR 501, from Anton Paar), with a 60 mm diameter and 1° titanium cone geometry.

The onset of flow instability is visually determined by the onset of intensity fluctuations in the channel. In order to set precisely the critical imposed flow, Q_{c}, for which the flow becomes time-dependent, we measured the root-mean-square of the intensity fluctuations in time (I_{rms})

(1) |

At the onset of flow instabilities the Reynolds number (Re = ρUW/η()), where ρ is the fluid density, and η() the rate dependent shear viscosity, is small, ranging from 0.05 (for the highest polymer concentration) up to 1.7 (for the lowest concentration). Therefore, in our microfluidic flow experiments inertial effects can be disregarded.

(2) |

(3) |

N_{1} = 2[η() − η_{s}]λ()^{2}, | (4) |

Shear-rate dependence of the shear relaxation time of polymeric solutions has been poorly reported in the literature.^{35,36} Typically, the relaxation time of complex fluids has been measured solely within the linear viscoelastic regime, for which the relaxation time is constant by definition.^{10,28} The theoretical work of Fielding and Olmsted^{37} uses a shear-rate dependence for the relaxation time of wormlike micellar solutions, but the amount of experimental data is limited for these systems.

We determine here the shear-rate dependence of the relaxation time following the White–Metzner model (eqn (4)) from the ratio of the first normal-stress difference N_{1} to the total shear stress σ = η() as

(5) |

In Fig. 2a and b we show the steady shear flow curves (shear viscosity and N_{1}vs. shear rate) for the various polymer concentrations measured at T = 23 °C. In the dilute regime (c = 125 and 500 ppm) the shear viscosity remains nearly constant for the entire range of shear rates measured (which largely covers the range of shear rates explored in the serpentine flow experiments) and N_{1} displays a quadratic dependence on the shear rate, as predicted by the Oldroyd-B model.

As the PEO concentration increases the total shear viscosity (which takes into account both polymeric and solvent contributions, η = η_{p} + η_{s}) plateau (recovered for small shear rates) increases gradually. In addition, the fluid response becomes non linear. The degree of shear thinning for the shear viscosity becomes progressively more pronounced, and the slope of N_{1} as a function of shear rate in log–log coordinates becomes smaller than 2, and decreases progressively with the increase of polymer concentration.

In Fig. 2c we show the results obtained for the relaxation time using eqn (5). For c = 1000 ppm, we find that the relaxation time is nearly insensitive to the shear rate, λ ≃ 6 ms (within the experimentally accessible range of shear rates). This is the smallest relaxation time that we can resolve experimentally with our protocol with reasonable accuracy. For c < 1000 ppm experimental uncertainties in N_{1} and (η() − η_{s}) lead to considerable unphysical fluctuations of the relaxation time. Within the dilute regime the relaxation time is theoretically independent of polymer concentration.^{22} Therefore, we use for c = 125 and 500 ppm λ = 6 ms, the value obtained experimentally for c = 1000 ppm. Considering the important variability reported between different batches of nominally identical PEO,^{39} this value is in reasonable agreement with previous findings using similar PEO solutions.^{14}

The coloured lines in Fig. 2 are fits of the White–Metzner model, which have been obtained by simultaneously fitting both η() and λ() for solutions with polymer concentration above c*. The model quantitatively captures the rheological response of PEO solutions in the semidilute regime, c ≥ 1000 ppm, for both shear viscosity and relaxation time. The rheological parameters obtained from the fits are shown in Table 1. As we have pointed out c = 1000 ppm is at the limit of the resolution of our experimental methodology. For this concentration it is not possible to measure shear thinning for the relaxation time, and we thus prefer not to fit the White–Metzner model in this case. For c = 1000 ppm, we include in Table 1 the measured value of λ_{0}, and the exponent n that has been obtained using a power-law fit to the shear viscosity.

c (ppm) | c/c* |
η
_{p,0} (mPa s) |
λ
_{0} (s) |
Λ (s) | n (−) |
---|---|---|---|---|---|

125 | 0.23 | 0.3 | — | — | — |

500 | 0.91 | 1.7 | — | — | — |

1000 | 1.83 | 4.8 | 0.006 | — | 0.88 |

2000 | 3.66 | 15 | 0.017 | 0.10 | 0.81 |

3500 | 6.40 | 56 | 0.05 | 0.50 | 0.75 |

5000 | 9.14 | 158 | 0.09 | 0.50 | 0.66 |

7500 | 13.71 | 498 | 0.15 | 0.80 | 0.60 |

The zero shear rate relaxation time λ_{0} increases with polymer concentration. Besides, for c > 1000 ppm the relaxation time displays a non-negligible shear-thinning behaviour (captured by the n exponent in the White–Metzner model). This shear-thinning behaviour becomes more significant as polymer concentration is increased. Since the degree of shear thinning predicted by the White–Metzner model for the first normal stress difference is more pronounced than the shear thinning for the shear stress, this model predicts a shear-thinning behaviour also for the relaxation time (determined viaeqn (5)), as observed in our measurements.

In Fig. 3a we summarize the results obtained for the polymer shear viscosity in the limit of small shear rates (η_{p,0}), for the whole set of solutions. Tirtaatmadja et al.^{40} showed that water and mixtures of water and glycerol are good solvents for PEO, and determined experimentally the solvent quality, ν = 0.55 (which lies between the θ-solvent, ν = 0.5, and good solvent limits, ν = 0.588^{22}). According to Rubinstein and Colby^{22} three different scaling regimes occur. For good solvents, in the dilute regime (c < c*) the polymer viscosity increases linearly with polymer concentration. In the semi-dilute unentangled regime (c* < c < c_{e}, where c_{e} is the entanglement concentration) the scaling is given by η_{p,0} ∼ c^{1/(3ν−1)}, which leads to η_{p,0} ∼ c^{1.54} upon substituting for the solvent quality. In the semi-dilute entangled regime (c > c_{e}) η_{p,0} ∼ c^{3/(3ν−1)}, and thus η_{p,0} ∼ c^{4.62}. We performed fits to the experimental data using the prescribed scaling exponents for a good solvent with ν = 0.55 (included in Fig. 3a), and identified three different concentration regimes. Although we are conscious that the validity of these fits is questionable (because we have only a few experimental points), the fits show that our experimental data follow approximately the predicted trends. At low concentrations (c = 125 and 500 ppm) we recover the linear scaling corresponding to the dilute regime; for intermediate polymer concentrations (1000 and 2000 ppm) the viscosity increases with a power-law exponent close to 1.54, and for the largest concentrations (c = 3500, 5000 and 7500 ppm) the slope increases further, suggesting that for this polymer solutions c_{e} ≃ 3500 ppm. These scaling results are in relatively good agreement with experimental data reported by Ebagninin et al.,^{27} Arnolds et al.,^{29} Del Giudice et al.,^{30} for comparable PEO solutions in good solvents (although larger values of the entanglement concentration, c_{e} ≃ 4500^{30} and 6600 ppm,^{27} were estimated).

Alternatively, the scaling laws for the relaxation time in good solvents are predicted to follow λ_{0} ∼ c^{(2−3ν)/(3ν−1)} in the semi-dilute unentangled regime (leading to λ_{0} ∼ c^{0.54} for ν = 0.55) and λ_{0} ∼ c^{3(1−ν)/(3ν−1)} in the semi-dilute entangled regime (λ_{0} ∼ c^{2.08} for ν = 0.55).^{22} These are in relatively good agreement with the experimental work of van Zanten et al.,^{23} Zell et al.,^{28} Del Giudice et al.^{30} for PEO solutions in good solvents. In Fig. 3b we show the dependence of the estimated zero-relaxation time λ_{0} as a function of polymer concentration, and include the corresponding fitting curves for completeness (although their relevance remains marginal because we have a few experimental points, and the uncertainty associated with the relaxation time estimate is significant).

Fig. 4 (a) Critical shear rate for the onset of elastic instabilities vs. polymer concentration measured in the microfluidic serpentine channels. (b) Critical Weissenberg number as a function of polymer concentration. The corrections for solvent viscosity and non-linear rheology are taken into account one at a time. (c) Critical effective Wi number vs. 1 − n, where n is the power-law exponent. Experimental results (green triangles) and 3D numerical simulations (red circles) obtained using the White–Metzner model are shown. The dashed red line corresponds to the situation of the absence of shear thinning obtained for the simulations using the UCM model.^{14} |

Interestingly, we observed in the course of our rheological measurements in cone-plate shear flows comparable destabilizing trends as those reported in Fig. 4 for serpentine flows. In cone-plate flow measurements one can detect the departure from a viscometric flow, by an apparent shear-thickening behaviour in the measured flow curve (which can be attributed to the onset of elastic flow instabilities, if Re < 1). We observed that the critical shear rate for the onset of apparent shear thickening in cone-plate flow followed the same trends with polymer concentration as the ones reported for serpentine flows in Fig. 4 (data shown in the ESI†). Even though in the cone-plate flow the inertial component is not negligible, we suggest that the enhanced stabilizing effect may not be restricted to serpentine flows.

It is known that solvent viscosity has a stabilizing effect on the flow of viscoelastic solutions^{15} as was shown experimentally in the serpentine microflow of dilute polymeric solutions.^{39} By increasing the polymer concentration, the relative contribution of the solvent viscosity to the total shear viscosity (η_{s}/η) decreases. For high polymer concentrations the effect of solvent viscosity becomes negligible (η_{s} ≪ η). In this concentration range, instead, the shear-thinning behaviour of polymer solutions becomes significant in the relevant range of shear rates.

To account for the influence of the solvent viscosity as well as the non-linear rheology of polymer solutions, we use the general expression of the instability criterion proposed by Pakdel and McKinley:^{13}

(6) |

Wi_{0} = λ_{0}. | (7) |

Wi_{0} > CM_{crit}, | (8) |

(9) |

(10) |

In Fig. 4b, we first plot Wi_{0} obtained for the onset of elastic instabilities, as a function of PEO concentration (squares). To test the influence of solvent viscosity on flow stability we then plot (upper triangles). In the dilute regime (c = 125 and 500 ppm), including the solvent viscosity correction rescales _{crit} into a dimensionless instability threshold nearly independent of the polymer concentration. The slight differences observed between these results and previous results using similar PEO solutions^{14} can be attributed to the fact that the Zimm relaxation was used in Zilz et al.^{14} as an estimate for the shear relaxation time.

We now take shear thinning into account by evaluating Wi_{eff} (side triangles). The effective Weissenberg number is independent of polymer concentration up to c = 2000 ppm. This indicates that Wi_{eff} correctly captures the effect of mild shear thinning on the instability threshold. But for larger polymer concentrations (c ≥ 3500 ppm), for which shear thinning becomes more important, the critical Weissenberg number displays a significant increase with concentration that persists, even after correcting for the shear-rate dependent rheology. This result suggests that the general Pakdel–McKinley criterion is not sufficient to explain the stabilization effect that we observe for shear-thinning polymeric fluids.

To investigate the stabilizing effect of shear thinning observed at high concentrations we compare our experimental results to 3D numerical simulations. This is shown in Fig. 4c. The critical Wi_{eff} number is plotted as a function of the shear-thinning exponent, 1 − n. The simulations were computed for the same geometrical conditions as used experimentally using a finite-volume method described in detail in Afonso et al.^{41} To highlight the effect of shear thinning on the instability threshold, we have included a dashed red line corresponding to the numerical value of the critical Wi_{eff} number obtained by Zilz et al.^{14} in the absence of shear thinning, using the upper-convected Maxwell model (UCM).

We observed previously in Zilz et al.^{14} that experiments and simulations were in very good agreement as far as scaling laws were concerned, but slight quantitative differences were observed in the numerical prefactors of Wi_{eff−crit}. This difference was attributed to the difficulty of determining precisely the relaxation times for such dilute polymer solutions and also to the limitations of the constitutive equations used. This also holds for experimental values obtained in the non-shear-thinning regime of the present study.

Our experimental results in Fig. 4c show that a clear departure from the non-shear-thinning scenario takes place for 1 − n > 0.2. Numerical simulations also show that the instability threshold for shear-thinning fluids is clearly higher than the threshold obtained for fluids with a constant viscosity (depicted by the dashed line in Fig. 4c). The instability threshold increases progressively with the degree of shear thinning. In both experiments and simulations shear thinning has a stabilizing effect on the serpentine flow, shifting the instability thresholds to progressively larger values, although the effect is less pronounced in the simulations. Overall, we observe a reasonably good qualitative agreement between experiments and simulations.

In order to evaluate the role of fluid elasticity compared to flow inertia we compute the elasticity number, El = Wi_{eff}/Re. At the onset of flow instabilities 0.5 < El < 87. Only for the most dilute solution is the elasticity number smaller than one, and inertia could thus have a non-negligible role. We performed experiments with the Newtonian solvent (with no elastic contribution) at comparable flow rates and verified that the flow remained stable. Therefore, the experiments were performed in a regime where inertia was negligible, and consequently inertial contributions are not at the origin of flow instabilities reported here.

Poole et al.^{42} showed numerically that in serpentine flows of viscoelastic fluids secondary flows develop, even in the inertialess regime, due to the curvature of the geometry and the streamwise first normal-stress difference (which can be interpreted as the viscoelastic equivalent of Dean vortices, for Newtonian fluids). These results have recently been supported by experimental evidence in the flow of polymer solutions.^{32,43} The strength of secondary flows increases linearly with the applied flow rate. Note that for dilute polymer solutions^{14} secondary flows can be disregarded in the stability analysis, since at the critical flow rates for the onset of elastic flow instabilities these are still very weak. However, we have shown here that for shear-thinning fluids the instability onset shifts towards higher applied flow rates. At these largest flow rates secondary flows may be relatively strong, and it remains unknown whether they have an impact on flow stability (for example by advecting flow perturbations towards regions with a smaller flow curvature). Nevertheless, experiments performed by Larson et al.^{16} in Taylor–Couette flows using shear-thinning polystyrene solutions also exhibited flow stabilization. Note that in the laminar large aspect ratio Taylor–Couette flows no secondary flows develop along the transverse direction (nor in cone-plate flows). We can therefore infer that secondary flows in serpentine channels, even though they cannot be disregarded, cannot be seen as the only mechanism responsible for flow stabilization.

The existence of second normal-stress difference, N_{2}, may also lead to a stabilizing effect of the flow (as predicted theoretically by Larson et al.^{16}). However, for polymeric solutions the magnitude of N_{2} is small compared to N_{1} (only beyond the crossover to the entanglement regime, c > c_{e}, the contribution of N_{2} may become more important^{10}). Doi and Edwards^{44} predicted that N_{2}/N_{1} = −2/7 at low shear rates for well entangled polymers. We estimated the influence of N_{2} using the Giesekus constitutive equation and the estimate of Doi and Edwards on the instability criterion, following the correction suggested by McKinley et al.^{15} However, by including the existence of N_{2} in the scaling criterion the onset of elastic instabilities is modified by less than 20%. Thus, we believe that the stabilization of the instability observed cannot be captured by a simple rescaling of the Pakdel–McKinley criterion (even when including shear thinning and N_{2}) suggesting a possible new instability mechanism.

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## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6sm00326e |

‡ Present address: Laboratoire Charles Coulomb UMR 5221 CNRS-UM, Université de Montpellier, Place Eugène Bataillon, 34095 Montpellier CEDEX 5, France. |

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