Article pubs.acs.org/Langmuir
Vorticity Banding in Biphasic Polymer Blends Sergio Caserta* and Stefano Guido Dipartimento di Ingegneria Chimica, Università di Napoli Federico II, UdR INSTM, P.le Tecchio, 80, 80125, Napoli, Italy ABSTRACT: Pattern formation under the action of flow is a subject of considerable scientific interest with applications going from microfluidics to granular materials. Here, we present a systematic investigation of shear-induced banding in confined biphasic liquid−liquid systems, i.e., formation of alternating regions of high and low volume fraction of droplets in a continuous phase (shear bands). This phenomenon is investigated in immiscible polymer blends sheared in a sliding parallel plate flow cell. Starting from a spatially uniform distribution of droplets, the formation of bands aligned along the flow direction is observed, eventually leading to an almost complete separation between droplet-rich and continuous phase regions. The initial band size is related to the gap dimension; the merging of bands and consequent spacing reduction has also been observed for long times. Shear banding is only observed when the viscosity of the dispersed phase is lower as compared to the continuous phase and in a limited range of the applied shear rate. Rheological measurements show that band formation is associated with a viscosity decrease with respect to the homogeneous case, thus implying that system microstructure is somehow evolving toward reduced viscous dissipation under flow.
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INTRODUCTION “Banded structures” of macroscopic dimensions induced by simple shear flow in many different types of soft matter systems are phenomena of increasing scientific interest.1−3 Flowinduced microstructure evolution leading to banded regions has been observed in several complex fluids, such as wormlike micellar solutions,4,5 rodlike virus suspensions,6 attractive emulsions,7,8 lyotropic liquid crystals,9 suspensions of rigid spherical particles,10−12 supramolecular polymer solutions,13 and granular materials.14 A quite popular type of flow used in these studies is shear flow,15 which is shown schematically in Figure 1 in the simplest situation of two parallel plates, the
attributed to the coexistence of regions with different shear rates at the same value of stress due to a flow instability.4 For similar systems in straight microchannel flow, the instability of the interface between the shear bands is associated with velocity modulations along the vorticity direction.16 By means of rheometry, the discipline pertaining flow, and deformation of soft matter, it was found that the rheological fingerprint of velocity gradient banding is a plateau in the shear stress vs shear rate region, where a high and a low stress branch phase-separate under flow. Formation of bands alternating along the flow direction has been reported in attractive emulsions, i.e., where droplets flocculate due to micellar depletion attractions, and it has been attributed to the interplay between flocs elasticity and wall effects. Migration of suspended particles along the velocity gradient direction can be related with non-Newtonian properties of the suspending fluid.17 As far as vorticity banding is concerned, which is observed in the case of rodlike virus suspensions where a transition between isotropic and liquid crystalline state is also observed, the situation is not as well understood.2,13,18 Vorticity banding was found in the relatively simpler situation of a mixture of two Newtonian, immiscible liquid phases with no interfacial agents, at low values of the viscosity ratio between the dispersed and the continuous phase, and in a given range of the applied shear rate;19 in this case band formation was also found to be associated with a viscosity decrease. Analytical bifurcation theory in terms of the Landau equation20,21 relates shear banding in granular fluids to a lower viscous dissipation. Phase-separated liquid−liquid systems, such as emulsions and polymer blends, enjoy a variety of applications, from
Figure 1. Parallel plates apparatus.
lower being translated with velocity V with respect to the upper along the x-axis, with the sample placed in between. The imposed rate of deformation or shear rate γ̇ is uniform and given by the ratio between V and the gap δ between the plates (actual shear flow devices include rotational geometries, too, such as concentric cylinders, where gradient of γ̇ is present). Depending on the material under investigation, band formation along either of the three axes of shear flow depicted in Figure 1 has been reported. For wormlike micellar solutions, velocity gradient banding has been observed, i.e., the bands alternate along the velocity gradient (y-axis in Figure 1), and it has been © 2012 American Chemical Society
Received: August 9, 2012 Revised: November 2, 2012 Published: November 7, 2012 16254
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Figure 2. Images of a 10% v/v sample at λ = 0.04 sheared at γ̇ = 0.5 s−1 between the sliding plates sketched and observed along the velocity gradient direction (y-axis). Strain increases from (a) to (h), and the value is indicated in the top left corner of each image. Sample thickness is 500 μm. Scale bar in a corresponds to 1000 μm. optical and rheometrical test here reported have been run at the constant temperature of 23 °C. The two polymers are mixed together by manual stirring and let to stand overnight to remove air bubbles before use. The so-obtained blends are quite stable against sedimentation due the high viscosity and the small density difference between the two polymers (0.08 g/cm3), ensuring that buoyancy effects are negligible. Rheological tests were carried out in a constantstress (Bohlin, CVO 120) and in a strain-controlled rheometer (Ares, TA Instruments) with cone-and-plate geometry. Both polymers exhibit Newtonian behavior in the shear rate range investigated (γ̇ = 0.05−5 s−1). By mixing silicone oils of different viscosity, it is possible to change the viscosity ratio between the droplets and the continuous phase of a blend. A preshearing at high γ̇ values (ca. 40 s−1) was applied before starting the actual testing of polymer blends in order to cancel out possible loading effects and homogenize the initial sample microstructure for sake of reproducibility. Parallel Plate Apparatus. A detailed description of the experimental technique, data analysis, and possible source of errors has been provided elsewhere;35,36 only the relevant features will be briefly summarized in this section. The samples were sheared in a flow cell consisting of two parallel plates made of optical glass (180 mm × 50 mm × 5 mm). Parallelism between the two plates is adjusted by exploiting the reflections of a laser beam from the plate surfaces through a set of micrometric rotary and tilting stages (the residual error is around 20 μm over a length of 150 mm); in the rheo-optical experiments here reported the gap width was set to 500 μm. Shear flow at a constant shear rate γ̇ is generated by translating one of the plates at a constant speed with respect to the other, via a computercontrolled motorized translating stage (Ludl) with micrometric precision settings. The motion was periodically reverted, due to the limited size of the plates, the extension of every run being 60 mm; the influence of the flow inversions was verified by changing the frequency of the inversions and hence the amplitude of the imposed square wave oscillations. A preshearing flow was applied as in rheological testing. All the experiments were performed at room temperature in a thermostated room. The sample is observed along the velocity gradient direction (y-axis in Figure 1) through a transmitted light microscope (Axioscope, Zeiss), and images are acquired by either an analogue standard CCD camera (KPME1, Hitachi) connected to a PC-hosted frame grabber or a high resolution CCD camera (C8484, Hamamatsu) with FireWire interface.37,38 The microscope itself is also mounted on a separate motorized translating stage (Newport), which is used for sample scanning in the horizontal plane. The individual images can be then combined together to form a large composite image. 3D optical sectioning is achieved by sample scanning along the y-axis too by means of a focus drive (Ludl). The operation of the shear device, including speed and strain settings and scanning control, is done remotely by an ad hoc
cosmetics to foodstuffs. The properties of these systems are strongly dependent on their microstructure, which in most cases is given by droplets of one phase dispersed in the other (continuous) one. The size and shape distribution of the dispersed-phase droplets are markedly affected by the flow conditions experienced during processing. Thus, flow-induced microstructure evolution has attracted much interest, starting from the pioneering work of Taylor in the early 1930s on the deformation of isolated droplets in an immiscible liquid undergoing shear flow.22,23 The two main mechanisms governing the action of flow on system microstructure are droplet deformation, which can eventually lead to breakup in smaller fragments, 24 and collision, 25 which may elicit coalescence into larger droplets.26 Since the shear flow field is confined between two solid surfaces, wall effects cannot be ruled out, even if the average droplet size is much smaller compared to the gap thickness. The effect of confinement on droplet deformation, breakup, and coalescence, being relevant for microfluidics applications, has been recently addressed in several studies.27−30 In particular, an intriguing droplet−string transition has been found in concentrated polymer blends at a viscosity ratio of one when the droplet size becomes comparable to the gap between the shearing surfaces.31,32 The strings, aligned in the flow direction, are formed as a result of drop alignment into pearl necklace structures followed by drop coalescence, possibly due to wall-induced distortion of the velocity field. The effect of viscosity ratio and volume fraction on structure development, including coalescence and breakup, has also been investigated.33,34 In this work, immiscible polymer blends, with volume fractions Φ ranging from 2.5% to 20% and viscosity ratios λ from 0.002 to 5, are subjected to shear flow in a sliding plate device such as the one in Figure 1. The vorticity banding dynamic has been quantitatively analyzed by image analysis techniques, a systematic rheological investigation of the viscosity decreasing is also provided.
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EXPERIMENTAL SECTION
Materials. In this work, the liquid−liquid model system selected for the experiments, widely used in the literature,24,25,27,28,31,32,34,35 is a blend of silicone oil (dispersed phase, from Dow Corning) and polybutene (continuous phase, from BP Chemicals), which are immiscible, transparent liquids at room temperature. All the rheo16255
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Figure 3. (a) The composite image at λ = 0.002 shows large droplets elongated along the flow direction. The enlarged view of the square frame is presented in (b) does not present bands along the vorticity direction. developed automated time-lapse software that collects images on hard drive for later analysis. Automated analysis of the y-stacks of images obtained from optical sectioning is performed by running a macro calling standard image analysis routines from the library of a commercial package (Image-Pro Plus, Media Cybernetics). The macro provides size and locations of the droplets within the scanned volume, which are then displayed as 3D reconstruction by scientific graphic software (SigmaPlot, Systat software). Scale factors from pixels to micrometers are obtained by imaging calibration stage micrometers (Graticules).
image was obtained by putting together adjacent images acquired through sample scanning in the x−z plane. The large droplets are quite elongated and would be unstable in unbounded shear flow. The stabilizing effects of the confining walls on drop breakup have been already observed.27,28,31,32 The absence of bands is confirmed by the enlargement in Figure 3b, where individual droplets are clearly visible, but no microstructure is observed. Data Analysis. To elucidate the three-dimensional structure of the shear bands, high-magnification optical sectioning was performed under static conditions after band formation. Image stacks have been processed to extract droplet size and location within the sample volume;34 droplet size appears to be maximal in the dispersed-phase-rich regions and to decrease to a minimum in the continuous-phase-rich ones. The higher droplet size in the droplet-rich bands is likely to be attributed to the enhanced coalescence rate at higher volume fraction, an effect which is well documented in the literature.33,34 To evaluate if band formation was associated with regions of high and low shear rate, velocity profiles were measured during flow both before and after the appearance of the banded structure, at different sample locations along the vorticity axis z.19 All the velocity profiles are linear going from the top to the bottom plate of the shearing device, and independent of the z location, thus showing that the shear flow field is homogeneous throughout the sample. In order to investigate the kinetics of band formation in more detail, a quantitative description of the bands is needed. This was done by fast Fourier transform (FFT) of images of the sheared sample, as shown in Figure 4. For each strain value at
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RESULTS Phenomenological Description. At viscosity ratios of 1 and above, the sample looks homogeneous throughout the shearing experiment (up to 100 000 strain units). So, it was quite unexpected when band formation was observed at first.19 The appearance of bands with drop-rich and -depleted regions alternating perpendicularly to the flow direction along the vorticity axis (i.e, z-axis in Figure 1) is noticed when blends of low viscosity ratio λ (below 1) are extensively sheared at a constant value of shear rate within a given range. An example is shown in Figure 2a−h, where a sequence of low-magnification micrographs taken during flow at increasing values of strain is presented for a 10% v/v blend with viscosity ratio of 0.04. Starting from a spatially uniform distribution of droplets (Figure 2a), band formation becomes more and more apparent as time goes on, until an almost complete separation between droplet-rich and droplet-devoid regions is reached, as shown in the subsequent pictures. Though rather deformed, the droplets are far from the breakup point, where the shape is no more stable under the applied shear flow and droplet fragmentation takes place. In fact, the only mechanisms acting to modify the microstructure, during our experiments, are drop deformation, flow-induced drop collision, and coalescence. It can be noticed that initial bandwidth (see the scale bar in Figure 2a) is comparable with gap thickness (δ = 500 μm). At high strain value (Figure 2e−h), a further kinetic evolution of the band structure has been observed, adjacent droplet-rich bands collapse above one of the depleted regions; this phenomenon leads to a reduced number of bands and increased bandwidth over long experiment times (Figure 2h). The morphology evolution depends on the dynamic equilibrium among the mechanisms of droplet coalescence, deformation, breakup, and hydrodynamic interactions together with band fusion. The detailed investigation of system morphology at very large times is beyond the scope of this work. Shear banding has been observed only in systems with low viscosity ratio and in a given range of shear rate. In the case of λ = 0.01 parallel pearl necklace structures extending along the vorticity direction are superimposed to the bands aligned in the flow direction and described so far.19 By further reducing the viscosity ratio down to λ = 0.002, no bands are observed, and large droplets aligned with flow direction are formed as a result of coalescence, as reported in Figure 3a, where a composite
Figure 4. Analysis of band formation kinetics. FFT profile along the direction of band orientation in the frequency domain (black arrow in the inset, where both the original image and part of the corresponding 2D FFT are reproduced).
least 20 images have been calculated, and the results are averaged. Being the bands vertically aligned in the original image (smaller inset), by plotting the amplitude of the FFT power spectrum along a horizontal line starting from the origin (continuous line in the larger inset) a major peak is found at the spatial frequency associated with band spacing. The peak 16256
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Rheological Measurements and Discussion. To further investigate the shear banding phenomenon, rheological tests were performed in a cone-and-plate rotational apparatus. Blend viscosity has been measured under extensive shearing (over 20 000 strain units γ) for different values of the imposed shear rate (γ̇ = 0.1−2 s−1), disperse phase volume fraction (Φ = 5−20%), and viscosity ratio (λ = 0.01−5.4). The blend with viscosity ratio λ > 1 showed constant viscosity, for every value of Φ and γ̇ investigated (data not shown for the sake of brevity); these systems have never shown the formation of banded structures during rheo-optical experiments. Blends with viscosity ratio λ < 1 showed a progressive decrease in the measured viscosity, until a plateau value is eventually reached at high strain values, with different kinetics and plateau values depending on the volume fraction Φ and the imposed shear rate γ̇. In Figure 7a, blend viscosity η (normalized with respect to the initial value η0) vs strain is reported for a blend having Φ = 20% and λ = 0.1, for different values of the imposed shear rate γ̇; in Figure 7b, the same measurement is reported for a blend having Φ = 5% and λ = 0.01. Data reported are measured with a stress-imposed rheometer; the results do not show significant changes operating with a strain-controlled instrument (data not reported). Furthermore, the observed trends are not significantly affected if, instead of a continuous shear flow, a periodic square wave oscillation is applied, so as to reproduce the flow conditions experienced by the sample in the rheo-optical sliding plate device. We also observed that during the flow, as the banding phenomenon proceeds and the viscosity decreases, the first normal stress difference N1 of the blend increases. In the inset in Figure 7c we report both viscosity and N1 as a function of the strain; for the case Φ = 20%, λ = 0.1, and γ̇ = 0.5 s−1. The first normal stress difference N1 (normalized with respect to the initial value N10) is shown on the right vertical axis (black triangles), while on the left vertical axis the viscosity is reported for comparison (white triangles, same data as in Figure 7a). The increase in the first normal stress difference can be possibly due to the stress induced by the droplets migrating along the z direction that accumulate mass in the droplet-rich bands, thereby causing reverse flow of matrix fluid in the droplet depleted bands and an extra pressure along the y direction. The systematic investigation and detailed analysis of the first normal difference evolution during the shear banding are beyond the scope of the present work and could be the object of future investigations. After the rheological tests, by carefully lifting the cone, a concentric banded structure analogous to the system in the
amplitude (normalized with respect to the image average gray level) is plotted as a function of strain in Figure 5. It can be
Figure 5. Amplitude of the maximum of the FFT profile as a function of strain. Band spacing in the inset is either the frequency of maximum amplitude at each strain (squares) or the characteristic length ξ (circles).
noticed that, following a delay period up to ca. 3000 strain units, where no significant peak can be identified, the amplitude exhibits a sigmoidal growth until a maximum is reached. In the inset of Figure 5 the band spacing is calculated as the (inverse) frequency of the maximum of the FFT spectrum (squared symbols); the spacing remains constant at a value around 800 μm, which is comparable to sample thickness (i.e., 500 μm). Similar results are obtained by calculating the characteristic length ξ = ∫ I(ki) dki/∫ kiI(ki) dki (circular symbols in the inset of Figure 5), where I(ki) is the value of the FFT spectrum at the frequency ki, where i is the direction in the Fourier domain perpendicular to band orientation in the real space.39 It should be mentioned that the FFT analysis of the banding kinetic has been truncated at strain value before the first appearance of the band fusion phenomena, the quantitative description of banding dynamic at high strain value being outside the scope of this work. The growth of the FFT peak amplitude in the range of strain investigated can be associated with a progressive band sharpening as droplets tend to concentrate within the darker stripes of the image. In fact, the variation of gray level along the z-axis (vorticity) is essentially sinusoidal, as shown by parts a and b of Figure 6 in a qualitative and quantitative way, respectively. The intensity-coded surface plot of the banded structure presented in Figure 6a is analyzed by plotting the corresponding x-averaged gray level as a function of z in Figure 6b and comparing the data with a sinusoidal fit (continuous line), which gives indeed a good representation of the experimental trend.
Figure 6. (a, b) Surface plot of the banded structure (image in the inset of Figure 4) and x-averaged gray level vs z-axis with a sinusoidal fit (continuous line). 16257
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Figure 7. Rheological testing in a cone-and-plate apparatus. Viscosity vs strain under extensive shearing at γ̇ = 0.1 s−1 (circles), 0.5 s−1 (triangles down), 1.0 s−1 (triangles up), and 2.0 s−1 (squares) measured in a constant-stress rheometer. (a) Φ = 20%, λ = 0.1, matrix phase Napvis 30, η = 93.25 Pa·s, dispersed phase SO mixture (83.4% SO 12 500, 16.6% SO 1000), η = 9.25 Pa·s. (b) Φ = 5%, λ= 0.01, matrix phase Napvis 30, η = 93.25 Pa·s, dispersed phase SO 1000, η = 1.065 Pa·s. (c) Φ = 5%, λ = 0.01, γ̇ = 0.5 s−1. On the left axis is reported the viscosity (white triangles); on the right axis is reported the first normal stress difference N1/N10 (black triangles). Same system as in (a). (d) The inset is a picture of the rheometer plate taken after gently raising the cone, so to leave most of the sample on the bottom plate, at the end of the test at γ̇ = 0.5, Φ = 20%, λ = 0.1. (e) A picture of the rheometer cone taken at the end of the band formation test, image acquired in reflected light microscopy (2.5×).
At this stage, by comparing the results from rheo-optical and rheometrical experiments, we can state that shear banding is observed at low values of the viscosity ratio, and for specific ranges of the imposed shear rate, furthermore the blend viscosity seems to decrease during the banding phenomenon, hence reducing the viscous dissipation. The droplet-rich and -depleted regions observed in the banded structure (see Figure 2 and ref 19) can be expected to correspond to high and low viscosity values, being the viscosity of an immiscible blend a growing function of the disperse phase volume fraction.22,23,40 Since shear rate was found to be homogeneous within the
rheo-optical experiments is observed, as shown in the inset in Figure 7d at viscosity ratios λ = 0.1, Φ = 20%, and γ̇ = 0.5 s−1 (on the contrary, the samples at λ ≥ 1 look structureless); similar structures have been observed after extensive shearing in a cone-and-plate rheometer of a suspension of rigid glass spheres in a viscoelastic polymer solution,12 where particles result packed in regular spaced aggregates. In Figure 7e, a picture of the rheometer cone at the end of a banding test is reported; the image is acquired in reflected light microscopy (2.5×), and the concentric banded structures are clearly visible. 16258
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Figure 8. Blend viscosity η as a function of the dispersed phase volume fraction, Φ, for different values of the shear rate γ̇ = 0.05 (crosses), 0.1 (circles), 0.5 (triangles down), 1.0 (triangles up), 2.0 (squares), 5.0 (stars), and different viscosity ratio λ = 5.4 (A), 1 (B), 0.1 (C), and 0.01 (D). The lines are polymonial fit of experimental data. (A) λ = 5.4, matrix phase Indopol H50, η = 10.9 Pa·s, dispersed phase SO 60 000, η = 59.0 Pa·s. (B) λ = 1.0, matrix phase Napvis 30, η = 93.25 Pa·s, dispersed phase SO mixture (76.9% SO 100 000, 23.1% SO 60 000), η = 92.91 Pa·s). (C) λ = 0.1 matrix phase Napvis 30, η = 93.25 Pa·s, dispersed phase SO mixture (83.4% SO 12 500, 16.6% SO 1000), η = 9.25 Pa·s. (D) λ = 0.011, matrix phase Napvis 30, η = 93.25 Pa·s, dispersed phase SO 1000, η = 1.065 Pa·s.
viscosity of the banded emulsion can result lower or higher of the unbanded emulsion, having uniform volume fraction, depending on the concavity of the viscosity vs concentration curve. In particular, if the second term dependence is greater than zero, as usually expected for ideal mixtures, blend demixing would result in a viscosity increase, while a concavity below zero would result in a reduced viscosity of the banded emulsion, which would result as a more stable configuration compared to the homogeneous system, due to the reduced viscous dissipation. The contribution to the global rheology of the fluid from the droplet-rich and -depleted regions could furthermore be influenced on a minor degree by the droplet
sample (at variance with gradient banding), this indicates the coexistence of regions with different values of shear stress (= ηγ̇) in vorticity banding. Such coexistence has been hypothesized in the case of vorticity banding in lyotropic liquid crystals.9 The banded emulsion viscosity, which is related to the global viscous dissipation of the system, should result in an appropriate average of the high and low viscosity regions, this value being equal to the viscosity of the initial homogeneous emulsion, if the dependence of the blend viscosity with the volume fraction can be considered linear. If second-order terms in the viscosity−concentration dependence are important, the 16259
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viscosity PDMS matrix (a system which is considered immiscible for practical purposes), droplet shrinkage takes place with time, paralleled by progressive increase of interfacial tension and droplet viscosity. Such a phenomenon has been attributed to selective migration of lower molecular weight chains of PB into the PDMS continuous phase. When the phases are reversed, an interdiffusion of the two polymers is still present, as analyzed in detail in the case of a 23.6 mPa·s PDMS droplet in a 11 Pa·s PB matrix.44
size; eventual partial miscibility of the phases should also be considered. In order to clarify this point, the viscosity of blends at different volume fraction Φ and viscosity ratio λ was measured under different values of the imposed constant shear rate γ̇. For each set of these three parameters (Φ, λ, γ̇) a fresh sample was loaded under the rheometer, presheared, and the measure averaged only for a few minutes, not long enough to allow the eventual formation of banded structures. In Figure 8A, the viscosity of a blend with λ = 5.4 is reported as a function of Φ; different symbols are relative to different values of the imposed shear rate γ̇, and the lines are polynomial fit of the experimental data. The viscosity appears to be substantially a linear function of the volume fraction Φ, for every value of the imposed shear rate γ̇. This system does not show shear banding or reduction of the viscosity during the flow. In Figure 8B, the same measurement is reported for a blend having λ = 1. Some differences appear among measurements run at different imposed shear rate due to the fact that different volume fractions show different shear thinning: mixtures with a volume fraction Φ < 15% can be considered Newtonian in the range of shear rate investigated, while blend having Φ = 30% show a viscosity reduction for γ̇ > 1 s−1. The viscosity has a linear growth with Φ for low concentration blend, while second-order contributions start to arise for Φ > 15%. Analogous conclusions could be also visible by plotting the same viscosity measurements as a function of the imposed shear rate, parametric in the volume fraction. In any case the concavity of each curve in Figure 8B can be still considered not negative, and the curvature can be neglected, within the experimental error, in agreement with the fact that these systems do not show shear banding during the rheooptical experiments. The differences from the ideal behavior became more evident at lower viscosity ratio; the lower the λ, the lower the value of Φ the curve deviate from the linearity. In the case of λ = 0.1 (Figure 8C) the concavity of the curves appears to change depending on the imposed shear rate, while the mixtures have constant viscosity for γ̇ < 0.1, and a stronger shear thinning the higher is the volume fraction Φ. This system has shown the formation of bands only for certain values of the imposed shear rate. In the case of γ̇ = 0.5 s−1 the blend shows a marked viscosity reduction, coupled with the formation of bands (Figure 7a), corresponding to a clearly negative curvature of the viscosity vs Φ (Figure 8C). In Figure 8D, the same measurement is reported for a blend having λ = 0.01; the viscosity appears roughly constant up to a volume fraction Φ = 15%, with a progressive thinning, in agreement with the other cases, but for higher volume fraction the viscosity shows a progressive decrement. In the rheo-optical experiments this fluid showed the formation of parallel pearl necklace structures extending along the vorticity direction superimposed to the bands aligned in the flow direction. By further lowering the viscosity ratio, no bands are visible during the rheo-optical experiments but elongated droplet (see the case λ = 0.002 in Figure 3). The behavior of systems having very low viscosity ratio (λ < 0.01) appears more complex in terms of both the morphology evolution and the rheological interpretation; a detailed analysis of these systems is outside the scope of this work. It is worth mentioning that for blends with very low viscosity ratios the diffusion of short chains fraction from the droplet to the matrix phase could not be completely negligible. It is known from previous studies41−43 that when an even highviscosity (around 105 Pa·s) PB droplet is immersed into a high-
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CONCLUSION In conclusion, we can state that in the banding phenomenon, upon inception of flow the initial structureless morphology evolves by formation of droplet clusters, which grow up to the size of the gap and connect together along the flow direction, thus creating the gap-dependent banded structure. Droplet clustering corresponds to the decreasing part of the viscosity vs strain plot (Figure 7), while the well-formed banded structure is observed once a viscosity plateau is reached. Once formed as a result of droplet collisions,45 the clusters that represent an intermediate stage of the shear banding phenomenon are stabilized by the associated reduction of viscous dissipation with respect to the homogeneous initial morphology, provided the blend and the flow conditions (i.e., λ and γ̇) are such as the viscosity vs concentration curvature is negative (Figure 8). The degree of actual separation among the bands, their speed of formation, and their dynamic evolution at long times (band fusion) depend on the dynamic equilibrium between the mechanisms acting to modify the system morphology, i.e., droplets coalescence, deformation, breakup, and hydrodynamic interaction, and to the value of the above-mentioned viscosity vs concentration curvature that represents the driving force leading to the observed reduction in blend viscosity. This proposed scenario of band formation is in line with recent perspectives on the mechanisms governing noise to order transitions,46 which share as common elements a randomizing effect (here diffusion due to droplet collisions) and a source of dissipation (here associated with viscous effects). The absence of banding at high viscosity ratios can be related to the lack of the viscosity-induced stabilizing effect (Figure 8). Hydrodynamic interactions may also play a role by lowering collision efficiency with increasing λ.47 The lack of band formation at the lowest viscosity ratio investigated (i.e., λ = 0.002) can be also attributed to the competing effect of coalescence which is higher the lower is the value of λ33,34 and can then elicit the formation of large, wall-stabilized droplets.27,28,31,32 The mutual diffusivity of the two polymer phases is also potentially involved. Within this framework, the fact that band formation is only found in a given range of shear rate can be explained by the opposing actions of shear-induced droplet diffusion tending to randomize the droplet spatial distribution at high values of γ̇48 and droplet coalescence, whose rate increases upon lowering γ̇.33,34 Shear banding in biphasic liquid systems appears as an intriguing novel phenomenology, being relevant to the study of the interplay between flow and microstructure, which is an important issue in the processing of these systems.
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AUTHOR INFORMATION
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[email protected]. 16260
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Author Contributions
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The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank Prof. G. Marrucci for useful discussions, Prof. N. Grizzuti and Dr. G. Filippone for the use of the straincontrolled rheometers, and A. Ferrara, A. D’Amato, and G. di Fonso for the experimental support during their undergraduate thesis.
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