Instabilities of micellar systems under homogeneous and non-homogeneous flow conditions
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- Méndez-Sánchez, A.F., López-González, M.R., Rolón-Garrido, V.H. et al. Rheol Acta (2003) 42: 56. doi:10.1007/s00397-002-0254-y
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The rheological behavior of a cetylpyridinium chloride 100 mmol l–1/sodium salicylate 60 mmol l–1 aqueous solution was studied in this work under homogeneous (cone and plate) and non-homogeneous flow conditions (vane-bob and capillary rheometers), respectively. Instabilities consistent with non-monotonic flow curves were observed in all cases and the solution exhibited similar behavior under the different flow conditions. Hysteresis and the sigmoidal flow curve suggested as characteristic of systems that show constitutive instabilities were observed when running cycles of increasing and decreasing stress or shear rate, respectively. This information, together with a detailed determination of steady states at shear stresses close to the onset of the instabilities, allowed one to show unequivocally that "top and bottom jumping" are the mechanisms to trigger the instabilities in this micellar system. It is shown in addition that there is not a true plateau region in between the "top and bottom jumping". Finally, the flow behavior beyond the upturn seemed to be unstable and was found accompanied by an apparent violation of the no-slip boundary condition.
Keywords.Micellar solutionsFlow instabilitiesTop and bottom jumpingSlipHomogeneous and non-homogeneous flow
Surfactants in solution are able to form molecular aggregates known as micelles, which can take different shapes depending on the concentration and ionic strength of the solution. Micellar solutions exhibit a complex rheological behavior that sometimes resembles that observed in high polymers. Typical phenomena displayed by micellar systems include flow instabilities such as shear banding and spurt.
Shear banding in micellar solutions may occur once a critical shear stress is reached (Berret et al. 1994; Berret 1997; Britton et al. 1999). Such a critical stress is usually associated with the onset of what seems to be a nearly constant stress plateau in the flow curve where the flow becomes unstable. In addition, under controlled stress conditions, similar to some linear polymer melts, micellar solutions may eventually spurt (Callaghan et al. 1996; Hernández-Acosta et al. 1999). In order to consider these phenomena, their overall flow behavior has been represented by a non-monotonic flow curve (Spenley et al. 1993). In contrast to the evidence available for linear polymer melts (Wang 1999), reported instabilities in micellar solutions have been considered to be constitutive in nature (Radulescu et al. 1999; Olmsted et al. 2000).
Several authors have tried to capture the different phenomena observed in micellar systems through theoretical frameworks (Spenley et al. 1996; Grand et al. 1997; Olmsted et al. 2000; Bautista et al. 2000). However, none of the available models is yet able to explain the whole range of possible phenomena in the non-linear rheology of micellar solutions.
Two issues that remain to be addressed are the influence of time on the appearance of flow instabilities in some micellar solutions (Hernández-Acosta et al. 1999) and the "top and bottom jumping" (Spenley et al. 1996) as the mechanism triggering the instabilities. Recently, Fischer (2000) reported time dependent structural changes in the flow of equimolar cetylpyridinium chloride/sodium salicylate (CPyCl/NaSal) micellar solutions, without hysteresis for upward and downward rate sweep experiments. Britton et al. (1999) reported apparent hysteresis above the critical shear rate for the onset of unstable flow for a 100 mmol l–1/60 mmol l–1 CPyCl/NaSal solution, but they were unable to verify such a result. The hysteresis was suggested as the result of the "top and bottom jumping" for increasing and decreasing shear rate, respectively. Grand et al. (1997) had previously reported that the 100/60 system does not show a simple "top-jumping" and pointed out the relevance of the time elapsed during the measurements.
Besides critical shear stress for the onset of instabilities, Hernández-Acosta et al. (1999) recently reported results about the effect of the residence time on the apparition of spurt instabilities in a micellar system under homogeneous and non-homogeneous flow (cone and plate and capillary rheometers, respectively). Moreover, an anomalous behavior consistent with a flow-induced phase change through capillaries was observed for very long residence times.
The study of the rheological behavior of micellar systems has been typically made under homogeneous flow conditions. This arises in part from the fact that a constitutive equation able to describe the whole bunch of phenomena observed in micellar systems is still lacking. However, the advantage of using homogeneous flow often disappears once the flow becomes unstable, since a mechanical instability changes the flow to a non-homogeneous one (Spenley et al. 1996; Britton and Callaghan 1997; Berret 1997).
In this work we analyze the flow behavior of the 100 mmol l–1/60 mmol l–1 CPyCl/NaSal system with a cone and plate rheometer. In addition, we include experiments under non-homogenous flow conditions using a capillary rheometer, as well as rotational relative rheometry by means of vane-bob geometry. The use of the capillary rheometer permits one to reach higher shear rates than in a cone and plate to investigate the upturn regions in non-monotonic flow curves associated with micellar solutions. On the other hand, possible slip at a rotating surface is eliminated as a variable when using the vane-bob geometry.
The collected experimental evidence in this work allowed us to show unequivocally the influence of time on the apparition of the instabilities, the existence of hysteresis in this solution as well as the "top and bottom jumping" for the onset of shear banding. Finally, the upturn in the flow curves was studied and found to be accompanied by what seems to be a violation of the no-slip boundary condition.
The studied system was cetylpyridinium chloride 100 mmol l–1/sodium salicylate 60 mmol l–1 in triple distilled water (CPyCl/NaSal) (Rehage and Hoffman 1988, 1991), which has been reported to develop shear bands and spurt under both homogeneous and non-homogeneous flow (Spenley et al. 1993; Callaghan et al. 1996; Britton et al. 1999). The CPyCl and NaSal (from Aldrich) had a purity of 98% and 99% respectively, and were used as received. Solutions were prepared in a dark room by dissolving the surfactant in water and then allowed to rest in dark glass bottles for one week in a water bath at 25 °C. Flow measurements were carried out at 25 °C, which is above the Kraft temperature for this system. Fresh samples were used in each experiment to prevent pre-shear history.
Experiments under homogeneous flow conditions were carried out by means of a Paar Physica rheometer (UDS 200) with the cone and plate geometry (25 mm in diameter and 2°). Dynamic oscillatory experiments as well as flow tests under controlled shear rate and stress were run. The linear viscoelastic behavior was consistent with that reported by other authors (Rehage and Hoffman 1991; Callaghan et al. 1996; Grand et al. 1997). The obtained longest relaxation time or Maxwell time and the plateau modulus were τR=1.7 s and GNo=36.6 Pa, respectively.
On the other hand, non-homogeneous flow experiments were also performed. The vane-bob geometry FL100 in a measuring cup of 22.5 mm was used in controlled torque and speed cycles. In addition, a pressure controlled capillary rheometer described elsewhere (Hernández-Acosta et al. 1999; de Vargas et al. 1993) was utilized. Capillaries of length (L) to diameter (D) ratio of 400 and D=0.12, 0.21, and 0.29 cm, respectively, were used. The pressure drop was measured with a Validyne differential pressure transducer located between the capillary ends. A mercury manometer was used to measure the injection pressure of the fluid. The flow rates were determined by collecting and measuring the ejected mass as a function of time.
At last, a simple but enlightening experiment was performed with the micellar solution in a glass beaker. The fluid was at rest while a thin glass plate was introduced and taken out at different speeds. The images were recorded by means of a video camera and the plate was visually examined to look for traces of the solution. The video can be seen in the Springer Verlag web page.
Results and analysis
Cone and plate flow
Three sets of flow experiments were carried out in the cone and plate rheometer. In the first set, steps in shear rate and stress were imposed while the system response (shear stress or shear rate, respectively) was monitored as a function of time. In the second set of experiments the overall flow behavior was analyzed with fast logarithmic ramps under controlled shear rate and stress conditions. The third set of experiments consisted of slow logarithmic shear rate and shear stress ramps. The purpose of this last set of experiments was to prevent overshoots and determine with accuracy the onset of the instability.
Steps and ramp under controlled shear rate conditions
When the fast shear rate ramp was imposed (a logarithmic one with 10 s between consecutive points, also in Fig. 1) a maximum in the flow curve close to 0.5 s–1 was clearly observed (~19 Pa). At higher shear rates, the shear stress values decreased and the flow curve displayed a plateau-like zone. The existence of such a plateau was considered by Grand et al. (1997) as evidence that the band selection mechanism is different from "top jumping". However, as we will show below, this is not the case.
Steps and ramp under controlled shear stress conditions
Note also from Fig. 2b that the region D recognized under controlled shear rate experiments can be distinguished here as the region in which the shear rate flattens at approximately 150 s. The agreement of the critical shear stress values for the onset of the instability or "top stress" (τT~19 Pa) from both controlled shear rate and stress experiments is remarkable (see the maximum in Fig. 1).
In contrast, the flow curve obtained with the fast shear stress ramp (a logarithmic one with 10 s between consecutive points, Fig. 2a) does not include a plateau region. It is seen that the unsteady step stress data for 19 Pa lie either above or below those obtained in the ramp. Apparently, in spite of the fact that the τT value has been surpassed, the time the system remains at a certain stress is not enough to reach neither steady states nor the unstable regions. In other words, in addition to a critical shear stress, a critical time is necessary to give rise to the instability. From Fig. 2b it seems that the instability is triggered at around 40 s after the critical shear stress is reached.
Lerouge et al. (2000) suggested that decay into the plateau represents a process of nucleation and growth of a shear induced phase for t>>τR. The results from the fast shear stress ramp discussed in the previous paragraph suggest that such experiment does not allow enough time for a complete transformation of the material and therefore the apparent plateau is not reached.
On the other hand, when the shear rate ramp was imposed in a fast way (not shown here), a similar behavior to that reported by Grand et al. (1997) (see their Fig. 3a) was observed. Namely, the flow curve collapsed into the apparent plateau once the critical residence time and shear stress were reached or surpassed. Grand and co-workers suggested this behavior as evidence that shear banding in this micellar system does not result from simple "top jumping". However, our results described in the previous paragraph show that, if stress overshoots are eliminated using slow shear rate ramps, true steady states can be reached even though the stress lies above the apparent plateau. It is interesting to observe that our τT is independent of the control parameter used in the experiments, shear rate or stress (see Fig. 3b). Thus, unlike the report by Grand and co-workers, the results in this work point towards the "top jumping" mechanism as the responsible for the shear banding in the 100/60 system.
The use of the cone and plate and concentric cylinder rheometers does not fully allow one to explore the possibility of "top and bottom jumping" as the mechanisms for the onset of the instabilities in micellar solutions. This is because the no-slip boundary condition at the moving surface is usually violated, and/or the solution leaves the gap once the critical shear stress and time are reached. By using vane-bob geometry we provide additional evidence that "top and bottom jumping" are the mechanism triggering the instabilities in this micellar system. Although this measuring geometry is not a standard one, it eliminates slip and ejection of the sample and provides useful qualitative results.
Several interesting characteristics can be observed in this figure. First, note that all the up and down data superpose in the stable flow regime (low shear rates). Besides, the upturn occurring in the flow curve at high shear rates is also observed. The most interesting feature of Fig. 5, however, is that the "top and bottom jumping" accompanied by hysteresis are clearly observed under both, controlled torque and speed, and that a true plateau in between does not really exist. At higher torque and speed values than those shown in the plots, the system started to exhibit the Weissenberg effect and finally the movement as a "whole" of the bulk of fluid in the cup. The differences in the up and down flow curves at the end of the unstable zone are likely due to fluid inertia. To our knowledge, evidence of the hysteresis cycles and the whole sigmoidal flow curve characteristic of a constitutive instability had not been previously reported for micellar systems.
Besides providing evidence for "top and bottom jumping" in this solution, Fig. 5 suggests the mechanism triggering the instability during the cycles. When increasing the torque or speed up to τT ("top jumping"), the fluid cannot longer sustain the deformation and shear banding occurs. As suggested by Lerouge et al. (2000) this could be the result of a flow-induced structure with small sub-bands aligned in the flow direction that makes it non-homogeneous. This in turn leads to a decrease in the average viscosity and a consequent increase in speed or decrease in torque depending on the controlled variable, torque or speed, respectively. On the other hand, when moving from high to low torque or speed up to τB ("bottom jumping"), flow-induced shear bands start to disappear, which now leads to an increase in the fluid viscosity and a resulting decrease in speed or increase in torque. This fast change in the fluid behavior is possible because any flow-induced structure has a short life-time. The time for stress relaxation after a shear rate step leading into what has been called the apparent plateau was found to be of the order of the Maxwell time (1.7 s). Thus, the process of nucleation and growth of any new phase takes a time of the order of 24 τR and is completely reversible. On the other hand, the "steady states" in what seems to be the plateau in Fig. 1 are probably due to stable shear bands induced when a stress overshoot overcomes τT due to a fast step in shear rate.
Finally, one important issue to point out in the experiments with the vane rheometer is the fact that the flow seemed to be unstable in the high shear rate branch. Although such behavior is not clearly observed during the up and down cycles, it is suggested by the fact that the flow superposition could not be performed, in contrast to that observed for the low shear rate branch. Evidently, such a proof of the flow stability cannot be given using a cone and plate rheometer since the high shear rate branch cannot be explored with this geometry. This result contrasts with the belief that the flow is stable in the high shear rate branch (see for example Spenley et al. 1996; Berret 1997). Additional proof of the stability of the flow in such a region should be provided, perhaps via optical methods.
Interfacial instability and capillary measurements
The unstable flow behavior described in the previous section has the signature of being constitutive in nature. There were indications during the experiments with the rotational rheometer, however, that the no-slip condition may be violated at high apparent shear rates, mainly those found beyond the upturn. A simple but enlighten visualization experiment is shown in the video (the video can be seen in the Springer Verlag web page). It is noted from the video that the fluid sticks to a glass plate at low shear rate, but it does not at higher ones. Although this experiment provides only qualitative information, this may be useful in making references to the origin of the different sort of instabilities occurring in micellar systems. The different flow regimes observed in a wide shear rate range in the 100/60 system were explored in detail using the capillary rheometer; the results are shown and discussed below.
In Fig. 6, several regions (I–V, delimited by arrows) that correspond to different flow regimes can be distinguished. Note that this flow curve is qualitatively similar to the one in Fig. 1. At the lowest shear rates, less than 0.4 s–1, (data mainly obtained with capillaries of big diameter) the fluid behavior is like a non-Newtonian shear-thinning one (I). Then, there is a region (II), starting at 12.5 Pa, (from 0.4 to 1.2 s–1) where the flow becomes unstable, showing small fluctuations in the pressure drop between capillary ends as well as in the shear rate. Note that changes in the pressure drop should not occur since the inlet pressure is kept constant in this kind of rheometer. It is noteworthy that the plot in Fig. 6 representing the injection pressure is monotonically growing, contrary to the flow curves of the different capillaries. The next region (III) in the flow curves, from 12.5 to 15.6 Pa (1.2 to approximately 4 s–1), corresponds to another stable flow regime characterized by a smaller shear-thinning index than in region I, and what seems to be a second local maximum, at 15.6 Pa, prior to the onset of a long plateau region (IV) where spurt flow is observed. The jump to the right branch of the flow curves during the spurt was not instantaneous and a transient of some minutes was observed.
It is difficult to know the precise shear rate at which the jump to the right branch of the flow curve took place for each capillary, since any small overshoot in the injection pressure caused immediate triggering of the jump. Note that some steady data obtained with the smallest capillary diameter lie in region IV, which suggests that the critical shear rate depends on the capillary diameter.
The critical shear stress (τc) for the plateau was 15.6 Pa, from which the ratio between the plateau modulus and the critical stress gives a value of τc/GNo=15.6 Pa/36.6 Pa=0.43, that is characteristic of materials prone to exhibit interfacial instabilities (Wang 1999).
Beyond the plateau there is a region (V), beginning at the upturn at high shear rates, in which the flow becomes almost stable again but there is a dependence of the flow curves on the capillary diameter. This fact suggests the presence of a positive slip velocity (Pérez-González et al. 1992; de Vargas et al. 1993). It is interesting to recall here the apparent lack of stability observed for this flow regime when using the vane rheometer. Perhaps this was not observed with the capillary because of the low sensibility of the utilized transducer.
Mair and Callaghan (1997) also studied the capillary flow of the same micellar system but their flow curve does not appear as detailed as the one in our Fig. 6. Note, however, the coincidence in the upturn at around 100 s–1 in both works (see their Fig. 5).
While the instability in the plateau, similar to that observed in the rotational rheometer, seems to have a constitutive origin, the behavior at shear rates beyond the upturn is consistent with slip flow. In this case, flow may be dominant over kinetics of the micelles, so their reactions are restricted. Under such conditions the behavior must be much like that in polymer melts and the slip phenomenon probably has the same origin, i.e., is interfacial in nature because of chain disentanglement (Wang 1999). The lack of adherence of the solution to a solid wall at a certain shear rates – but its adherence in others – supports the previous hypothesis (see the video).
Our conclusion in the above paragraph is in contrast to the hypothesis given by Mair and Callaghan (1997) of a constitutive instability for a similar solution. It is possible, however, that a combination of slip and shear banding is acting at the same time. For example, a nematic phase (see Berret et al. 1994) with parallel orientation to the capillary wall would give a diameter dependence similar to the one observed in the presence of positive slip (see Wissbrun 1981).
Despite the difference in the type of flow generated in rotational and capillary rheometers (homogeneous and non-homogenous), the coincidence in the shear rates where the instabilities take place in both rheometers is outstanding. Also the critical shear stresses are comparable. These facts suggest, in agreement with Berret (1997), Britton and Callaghan (1997), and Lerouge et al. (2000), that once the first instability is triggered in the cone and plate rheometer the flow becomes non-homogeneous, so the corresponding shear rate is only apparent. In this work shear banding in the cone and plate, whether shear rate or stress is controlled, occurs at very low shear rate and therefore it is possible that the flow in this rheometer is non-homogenous almost in the whole range of shear rates studied.
The cone and plate, vane-bob and capillary flow of micellar aqueous solutions of cetylpyridinium chloride 100 mmol l–1/sodium salicylate 60 mmol l–1 was studied in this work. Evidence was provided of the influence of the residence time on the onset of instabilities that take place at a critical shear stress. Hysteresis was observed and it was shown that "top and bottom jumping" are the mechanisms to trigger the instabilities in this micellar system. It was shown, in addition, that there is not a true plateau region in between the "top and bottom jumping". Finally, the upturn region in the flow curves seems to be unstable and accompanied by apparent slip.
This work was supported by CGPI-IPN (010565) and CONACYT (34971-U). A. F. M-S is SUPERA fellow. J.P-G and L. de V. are COFFA-EDI fellows. M. R. L-G and V. H. R-G had scholarships from PIFI-IPN and CONACYT. We wish to acknowledge to Prof. Jay D. Schieber for very useful discussions.