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Structure Development in Confined Polymer Blends: Steady-State Shear Flow and Relaxation Anja Vananroye, Peter Van Puyvelde,* and Paula Moldenaers Department of Chemical Engineering, Katholieke UniVersiteit LeuVen, W. de Croylaan 46, B-3001 LeuVen (HeVerlee), Belgium ReceiVed October 17, 2005. In Final Form: NoVember 30, 2005 In this work, the structure development in immiscible polymer blends in confined geometries is systematically investigated. Poly(dimethylsiloxane)/poly(isobutylene) blends with a droplet-matrix structure are subjected to simple shear flows. The confined environment is created by using a Linkam shearing cell in which the gap is systematically decreased to investigate the transition from “bulk” behavior toward “confined” behavior. Small-angle light scattering experiments in a confinement, which have not yet been reported in the literature, and also microscopy are used to observe the morphology development during steady-state shearing and relaxation. These experiments indicate that the size and relaxation of single droplets in a confined environment are still governed by the relations that describe the structure development in bulk situations. Yet, depending on the applied shear rates and blend concentrations, the droplets organize in superstructures such as pearl necklaces or extended superstrings in a single layer between the plates. These structures are stable under flow. To observe a single layer, a critical ratio of droplet size to gap spacing is required, but this ratio is clearly below the one already reported in the literature.
Introduction Since the initial research of Taylor1,2 on emulsions in the 1930s, the structure development in immiscible polymer blends in different types of flow fields has been thoroughly explored. Especially in the case of dilute blends, consisting of Newtonian components and subjected to a shear flow field, major progress has been made in understanding the rheological properties and morphology development of the blends. Most of the theoretical and experimental results have been summarized in several reviews.3-7 In the case of Newtonian components, two dimensionless numbers govern the blend behavior: the capillary number Ca (Ca ) ηmRγ˘ /R, where ηm, R, γ˘ , and R denote, respectively, the matrix viscosity, the droplet radius, the shear rate, and the interfacial tension) and the viscosity ratio p (p ) ηd/ηm, with ηd the droplet viscosity). When Ca exceeds a critical value Cacrit, the droplets will deform irreversibly under flow until, eventually, breakup occurs. For blends consisting of Newtonian components, Cacrit only depends on the type of flow and the viscosity ratio p. The relation Cacrit(p) was carefully mapped out by Grace,8 and a fitting relation was proposed by de Bruijn.9 Several theories have been developed to express the deformation of Newtonian droplets in bulk flows.1-7 Small deformation theories assume that the shape of the droplet is close to spherical, which applies when Ca , 1. Under these circumstances, the droplet deformation is usually expressed by means of the deformation parameter D:
D)
L-B L+B
(1)
where L and B are the major and minor axes of the droplet in the velocity-velocity gradient plane as can be seen in Figure 1. * To whom correspondence should be +32-16-32.23.57. Fax: +32-16-32.29.91.
[email protected].
addressed. E-mail:
Phone: Peter.
(1) Taylor, G. I. Proc. R. Soc. London, A 1932, 138, 41-48. (2) Taylor, G. I. Proc. R. Soc. London, A 1934, 146, 501-523. (3) Rallison, J. M. Annu. ReV. Fluid Mech. 1984, 16, 45-66. (4) Stone, H. A. Annu. ReV. Fluid Mech. 1994, 26, 65-102. (5) Ottino, J. M.; De Roussel, P.; Hansen, S.; Khakhar, D. V. AdV. Chem. Eng. 1999, 25, 105-204. (6) Tucker, C. L.; Moldenaers, P. Annu. ReV. Fluid Mech. 2002, 34, 177-210. (7) Guido, S.; Greco, F. Rheol. ReV. 2004, 99-142. (8) Grace, H. P. Chem. Eng. Commun. 1982, 14, 225-277. (9) De Bruijn, R. A. Ph.D. Thesis, Eindhoven University of Technology, 1989.
Figure 1. Scheme of a deformed drop with the principal distances and orientation in shear flow: (a) top view (observation plane); (b) side view.
Taylor1 has shown that, for nearly spherical droplets, the deformation D is proportional to the capillary number Ca. For higher capillary numbers, slender-body theories are used.10-12 The cross-section of the droplet is assumed to be circular (B ) W), and the deformation is given by the aspect ratio rp:
rp ) L/B
(2)
Also, a few phenomenological models are developed to describe droplet deformation. For example, Maffettone and Minale proposed a simple phenomenological model (MM model) to predict the shape of ellipsoidal droplets.13 This model is capable of describing the evolution of the three axes of a droplet in an arbitrary flow field up to approximately Ca ) 0.4, with conservation of volume at any deformation.6,13 Despite these successes in understanding the simplest case of Newtonian components in a simple shear flow, real-life polymer blends are still poorly understood because of additional complexities. These include viscoelastic components, concentrated systems, and complex flow histories. Also, a recent trend that (10) Taylor, G. I. Proceedings of the 11th International Congress on Applied Mechanics, Munich, 1964. (11) Hinch, E. J.; Acrivos, A. J. Fluid Mech. 1980, 98, 305-328. (12) Khakhar, D. V.; Ottino, J. M. J. Fluid Mech. 1986, 166, 265-285. (13) Maffettone, P. L.; Minale, M. J. Non-Newtonian Fluid Mech. 1998, 78, 227-241.
10.1021/la0527893 CCC: $33.50 © 2006 American Chemical Society Published on Web 01/10/2006
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is emerging in the process industries is toward miniaturization, a tendency that is also manifested in microscale processing of polymers.14,15 For multiphase materials such as polymer blends, miniaturization implies that droplet-wall interactions will become important, since the size of the dispersed phase becomes comparable to that of the flow geometry. This feature could drastically influence the morphological changes; therefore, the rules that govern macroscale processes cannot be extrapolated to microscale processes in a straightforward manner. Although a complete physical picture of the various morphological processes in confinements is still lacking, some interesting applications have already been presented. For example, Link et al.16 demonstrated that, with microfluidic technology, it is possible to design dispersions with controlled volume fractions and droplet sizes. In their setups, rather simple microfluidic channel shapes (i.e., flow past obstacles and T junctions) were used to passively break large drops into precisely controlled daughter drops. Another application, reported by Utada et al.,17 is to generate monodisperse double emulsions from a microcapillary device. Some remarkable phenomena, which start to contribute to a more general understanding of the processes, have already been reported. Migler18 observed a transition from dispersed droplets to a stringlike state in confined concentrated polymer blends (28 wt % dispersed phase) with p equal to 1. His experimental results indicate that a critical ratio of 0.5 for the droplet size to the gap is required to observe this transition. The strings in these emulsions are formed by coalescence of droplets, and they are stable due to a suppression of the Rayleigh instabilities, caused by finite size effects as well as by shear flow.19,20 Pathak et al.21 investigated the effects of the mixture composition and confinement on the microstructure in the transition zone between bulklike behavior and string formation for p equal to 1. Their results could be presented in a morphology diagram for confined emulsions of poly(dimethylsiloxane) (PDMS) and poly(isobutylene) (PIB) in the parameter space of mass fraction and shear rate. In addition to the formation of strings, they also reported the organization of particles into layers, depending on the concentration and the applied shear rate. The number of layers decreases from two to one as the droplets grow larger due to coalescence, and therefore, the ratio of the droplet size to the gap spacing exceeds a critical value of 0.5 for semiconcentrated blends. The physics behind this layered structure are not yet completely understood. Pathak et al.22 also investigated droplet deformation under confinement in a system with a viscosity ratio of unity and with a 9.7% mass fraction of the dispersed phase. It was shown that confinement allows droplets to exist beyond the critical capillary number for breakup, giving rise to stringlike structures with high aspect ratios. The aspect ratio of a droplet with dimensions below the critical value seemed unaffected. Here, the structure development in immiscible PIB/PDMS blends in confined geometries is investigated, using optical microscopy and small-angle light scattering (SALS). With microscopy, a local view of the blend morphology is obtained, whereas the SALS patterns reflect a more global view of the (14) Stone, H. A.; Kim, S. AIChE J. 2001, 47, 1250-1254. (15) Stone, H. A. Annu. ReV. Fluid Mech. 2004, 36, 381-411. (16) Link, D. R.; Anna, S. L.; Weitz, D. A.; Stone, H. A. Phys. ReV. Lett. 2004, 92, 0545031-0545034. (17) Utada, A. S.; Lorenceau, E.; Link, D. R.; Kaplan, P. D.; Stone, H. A.; Weitz, D. A. Science 2005, 308, 537-541. (18) Migler, K. B. Phys. ReV. Lett. 2001, 86, 1023-1026. (19) Son, Y.; Martys, N. S.; Hagedorn, J. G.; Migler, K. B. Macromolecules 2003, 36, 5825-5833. (20) Hagedorn, J. G.; Martys, N. S.; Douglas, J. F. Phys. ReV. E 2004, 69, 0563121-05631218. (21) Pathak, J. A.; Davis, M. C.; Hudson, S. D.; Migler, K. B. J. Colloid Interface Sci. 2002, 255, 391-402. (22) Pathak, J. A.; Migler, K. B. Langmuir 2003, 19, 8667-8674.
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blend structure. The influence of blend concentration and confinement (quantified as the ratio of the droplet size to the gap spacing) on droplet deformation and blend morphology is systematically investigated during steady-state shear flows, and the results are compared to the work of Migler and Pathak. In addition, the relaxation mechanism of confined droplets after cessation of shear is studied, as well as the mean steady-state droplet size. The experimental results are discussed in comparison with the existing bulk theories. Materials and Methods Materials. The blends used in this work consist of PIB droplets (Parapol 1300 from Exxon) embedded in PDMS (Rhodorsil 200000 from Rhone-Poulenc). The zero-shear viscosities of PIB and PDMS at 25 °C are 93 and 196 Pa‚s, respectively, resulting in a viscosity ratio p of 0.47. The critical capillary number Cacrit for this p, calculated with the correlation of de Bruijn,9 is 0.46. The viscosity of both pure components is constant up to the highest shear rates used here, and as elasticity effects are negligible, the pure components behave as Newtonian fluids under the measurement conditions.23 The density difference between PIB (FPIB ) 890 kg/m3 at 20 °C) and PDMS (FPDMS ) 970 kg/m3 at 20 °C) is small enough to neglect gravitational effects.24 The interfacial tension R for the PIB/PDMS system, measured by Sigillo et al.25 using different techniques, is reported to be 2.8 mN/m. The rheological and morphological behavior of this blend under bulk conditions has been thoroughly investigated,23,26 and will serve as a reference to compare the experiments in confined situations. Three compositions (1%, 5%, and 10% mass fractions of PIB) are chosen to study the influence of concentration in confined flows. Mixtures are prepared by manually stirring suitable amounts of the components with a spatula until a white and creamy appearance is observed. The entrapped air bubbles are removed in a vacuum oven at room temperature over 12 h. This procedure of blend preparation has been used previously and proved to be adequate.23,26-27 Methods. The confined environment is created in a Linkam shearing cell (CSS 450 from Linkam Scientific Instruments) consisting of two parallel quartz plates. A stepper motor is used to decrease the gap to investigate the transition from “bulk” behavior toward “confined” behavior. Prior to each experiment, the gap spacing and the parallelism of the plates were carefully checked by performing transmitted light intensity measurements on colored solutions at different gaps. Using this procedure, the gap setting has an accuracy of (5 µm. To characterize the affinity of PIB and PDMS for the plates, the contact angle of droplets of PIB and PDMS on a quartz plate in contact with air is measured with a CAM 200 (KSV Instruments). Contact angles of, respectively, 22.5° and 6.0° are found for PIB and PDMS droplets. Both microscopy and SALS are used to detect the morphology development during steady-state shear flow and to analyze droplet relaxation after cessation of flow. Microscopically, it is possible to visualize objects above micrometer size, whereas SALS can be used to probe structures on the micrometer and submicrometer length scales. Microscopic observations are made by placing the cell on an optical microscopy stage (Leitz Laborlux 12 Pol S). All images are acquired in the velocity-vorticity plane (see Figure 1) with a digital camera (Hamamatsu C4742-95) and analyzed using image analysis software (Scion Image). To acquire SALS images, He-Ne laser light (λ ) 633 nm) is sent through the cell, and the scattered light is visualized on a semitransparent screen parallel to the velocityvorticity plane. The scattering patterns are recorded with a digital camera (TM-1300 from Pulnix) connected to a digital frame grabber (23) Vinckier, I.; Moldenaers, P.; Mewis, J. J. Rheol. 1996, 40, 613-631. (24) Minale, M.; Moldenaers, P.; Mewis, J. Macromolecules 1997, 30, 54705475. (25) Sigillo, I.; di Santo, L.; Guido, S.; Grizzuti, N. Polym. Eng. Sci. 1997, 37, 1540-1549. (26) Yang, H.; Zhang, H. J.; Moldenaers, P.; Mewis, J. Polymer 1998, 39, 5731-5737. (27) Takahashi, Y.; Kurashima, N.; Noda, I.; Doi, M. J. Rheol. 1994, 38, 699-712.
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(TCI-Digital SE from Coreco) and analyzed with in-house-developed software (KULeuven SalsSoftware). All experiments are performed at a temperature of 25 ( 1 °C. After the sample is loaded, the initial droplet radius R is microscopically checked to be sure it is small enough to ensure that, for the initial shear rate subjected to the sample (0.38 s-1), coalescence is the only mechanism to occur. At this shear rate, microscopy images are recorded every 500 strain unitssdefined as γ˘ t, where t is the shearing timesto check for morphological changes. In this manner, it can be concluded that shearing for 5000 strain units provides adequate time for the blends under investigation to reach their steadystate morphology. Shear rates between 0.38 and 3 s-1 are applied to the system. The droplet deformation will be expressed by means of the aspect ratio rp. From the microscopy images under flow, only the axis W in the vorticity direction can be directly determined, as can be seen in Figure 1. The principal axis L, however, is oriented at an angle θ with respect to the flow direction, and therefore, only its projection Lp is seen in the observation plane. Equation 3 expresses L2p ) B2 + L2 cos2 θ - B2 cos2 θ
(3)
Lp as a function of L, B, and θ for an ellipsoid. The orientation angle, necessary to determine L and B, will be calculated from the analysis of Maffettone and Minale for simple shear flow:13
( )
f1 1 θ ) arctan 2 Ca
(4)
rates are applied to the system at a 100 µm gap spacing, smaller droplets will be present due to breakup, causing the degree of confinement to decrease.
f1 is a function depending only on the viscosity ratio p: f1 )
40(p + 1) (2p + 3)(19p + 16)
Figure 2. Steady-state morphology at (a) 100 µm and (b) 40 µm at _ γ ) 0.38 s-1. Inset: SALS pattern at 40 µm (1 wt % PIB in PDMS).
(5)
According to Maffettone and Minale,13 the orientation angle of droplets under bulk conditions is well predicted by eq 4. Moreover, confined single-droplet experiments in a counterrotating parallel plate device28 confirm that the prediction of the MM model for the orientation angle under simple shear flow is still acceptable as long as the ratio of the droplet diameter to the gap is 5, where H is the distance from the droplet’s mass center to the closest wall. For droplets near the midplane between the walls with a gap of 100 µm, this condition is satisfied since the droplets have a mean diameter of 12 µm at a shear rate of 0.38 s-1. When higher shear (28) Vananroye, A.; Van Puyvelde, P.; Moldenaers, P. To be submitted for publication. (29) Guido, S.; Simeone, M. J. Fluid Mech. 1998, 357, 1-20.
The average droplet diameter at a gap of 40 µm is approximately 10 µm; therefore, the average droplet dimension in the velocity gradient direction Lv (see Figure 1) is still small enough to allow for two or more layers of droplets to exist. However, as can be seen in Figure 2b, droplets are clearly all in focus during shear at 0.38 s-1, indicating that they are positioned in a single layer between the plates. Some alignment of droplets, the so-called pearl necklace structures, can be seen at shear rates of 0.38 and 0.75 s-1 at 40 µm. Also, the SALS pattern after shear at 0.38 s-1, shown in Figure 2b, is relatively isotropic, but some secondary streaks are weakly visible, indicating that some arrangement is present under these conditions. At shear rates of 1.5 s-1 and higher, the arrangement of droplets on pearl necklaces and the ordering in a single plane disappear again, due to breakup of the droplets and the decrease in the degree of confinement. Similar observations were made for the inverse blend (PDMS in PIB) at 36 µm by Pathak et al.21 In their research, PDMS and PIB both with a viscosity of 10 Pa‚s at 25 °C were used, and due to this low viscosity of the matrix, the influence of confinement was observed at shear rates up to 8 s-1. There, no pearl necklace structures were observed for the 1% blend, but instead, they reported a single-layer disordered microstructure. No information about the droplet size of the 1% blend was given. For concentrated blends, Pathak observed the transition to a single layer for 2R/d ≈ 0.5. In our case, however, for the dilute blend, the single layer is even seen for 2R/d ≈ 0.25. Figure 3 shows the droplet size as a function of shear rate for a 1% blend at different degrees of confinement. At each gap size, the experiments were performed with increasing shear rate. Error bars represent the standard deviation on the mean droplet radius. The average was taken over approximately 40 droplets at each shear rate and gap spacing. The breakup line, calculated from the critical capillary number,8,9 and the coalescence line, based on the partially mobile interface model (PMI model), have been
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Figure 3. Steady-state droplet size as a function of shear rate for three plate spacings (1 wt % PIB in PDMS).
added for comparison. In the case of the PMI model, the droplet radius Rcoal is expressed as30
Rcoal )
(
)( )
4 hcrit x3 p
2/5
ηmγ˘ R
-3/5
(6)
in which hcrit is the critical film thickness of the matrix fluid layer between approaching droplets. Equation 6 is fitted through the data point at a gap of 100 µm at the lowest shear rate (0.38 s-1), which is known to be generated through coalescence. From this fit, a value for hcrit of 75 nm is obtained. Between the breakup and coalescence curves, a hysteresis region is present where neither breakup nor coalescence will occur. At a specific shear rate, the critical shear rate γ˘ crit, the two curves cross, and beyond this crossover point, dynamic equilibrium can be achieved between the two phenomena. For the experiments performed here, a value of 5.6 s-1 is found, which is in agreement with the results of Minale et al.31 From Figure 3, it is clear that the mean droplet radii in confined (30 and 40 µm) and bulk (represented by the results at 100 µm) situations are still comparable to each other, so under the present conditions, no difference in droplet size was seen. Hence, the steady-state droplet sizes of single droplets in a 1% blend are still governed by the relations that describe the structure development in nonconfined situations. To assess the importance of concentration on the structure development in the blend, the same experiments have been performed on a 5% blend. As an example, Figure 4 compares the morphology and droplet size for the 5% blend at gaps of 100 and 40 µm after the initial shear rate of 0.38 s-1. The same phenomena as reported for the 1% blend are observed, although more pronounced. At a gap spacing of 100 µm, no arrangement in a single layer or in pearl necklaces can be seen, so a bulklike morphology is still present. At 40 µm, however, again a singlelayer structure and clear pearl necklace arrangements in the flow direction are seen. The average droplet diameter in this case is 20 µm, meaning that the average droplet dimension in the velocity gradient direction Lv is too large for two layers of droplets to exist. This is in agreement with the observed single-layer structure as was also reported by Pathak et al.21 The pearl necklaces in our experiments are recorded as a stable morphology during flow since no change in the overall blend morphology was seen during the final 2000 strain units of shearing. As can be seen in the microscopy image in Figure 4b, the distance z between two subsequent droplets in a necklace, measured as the distance in the flow direction between their mass centers, is approximately (30) Chesters, A. K. Trans. Inst. Chem. Eng. 1991, 69, 259-270. (31) Minale, M.; Mewis, J.; Moldenaers, P. AIChE J. 1998, 44, 943-950.
Figure 4. Steady-state morphology at (a) 100 µm and (b) 40 µm at γ˘ ) 0.38 s-1. Inset: SALS pattern at 40 µm (5 wt % PIB in PDMS).
constant. For example, a mean value for z of 42 µm is obtained during and after shearing at 0.38 s-1. Not only the individual droplets on a necklace but also the pearl necklaces themselves are ordered, resulting in an approximately constant distance y in the vorticity direction between two adjacent pearl necklaces. This internecklace distance is large compared to the average droplet size and compared to the interdroplet distance. From the microscopy image, an average value of 85 µm can be measured. It needs to be stressed that this organization of droplets is not a local phenomenon. This is proven by examining the SALS patterns, taken under the same conditions as the microscopy images. Whereas the SALS patterns of blends under bulk conditions normally become isotropic after steady-state shear flow, the present patterns show pronounced anisotropy. Perpendicular to the flow direction, secondary streaks, remaining visible even after the flow is stopped, can be seen. The presence of such secondary streaks in the pattern is due to the aligned structure of the droplets in the necklaces. Similar patterns have been reported by Mewis et al.,32 when filaments in a blend undergo fragmentation into an array of droplets due to Rayleigh instabilities. Mewis et al. used the Fraunhofer diffraction theory to analyze the SALS patterns. The interparticle distance z of droplets in a necklace can be calculated from the SALS patterns as32
z)
λ ηm sin φ
(7)
with λ the wavelength of the incident light, ηm the medium viscosity, and φ the polar scattering angle. For the SALS pattern in Figure 4b, this results in an interdroplet distance z of 37 µm, a value that is in good agreement with that measured on the microscopy image (42 µm). It is expected that the SALS pattern gives a more representative result, since it covers a more average picture of the blend morphology (scattering volume ∼0.15 mm3) than the microscopy image (0.27 × 0.20 mm2). The distance in the vorticity direction between two pearl necklaces y is too large (32) Mewis, J.; Yang, H.; Van Puyvelde, P.; Moldenaers, P.; Walker, L. M. Chem. Eng. Sci. 1998, 53, 2231-2239.
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Figure 5. Steady-state droplet size as a function of shear rate for two plate spacings (5 wt % PIB in PDMS).
to be seen on the SALS patterns; it would occur at an angle of 0.31°, which is beyond the scope of an SALS experiment. Figure 5 shows the mean droplet size as a function of shear rate for the 5% blend at two gap spacings: 40 and 100 µm. As can be seen in the figure, the experimental results at the two gaps still superimpose. The coalescence line is fitted by passing the curve through the lowest shear rate data point at 100 µm, resulting in a value for hcrit of 250 nm. This value is larger than that calculated for the 1% blend (hcrit ) 75 nm), which is in agreement with the findings of Minale et al.,31 who showed that an increase in the volume fraction of the dispersed phase results in an increase of hcrit. For semiconcentrated blends (as is the case here), hcrit loses its physical meaning, and it becomes a curve-fitting parameter. Due to this larger value, the intersection between the breakup line and the coalescence line is shifted to a lower shear rate (γ˘ crit ) 1.65 s-1). Above a shear rate of 2 s-1, it is no longer possible to perform a reliable analysis of the microscopy images of a 5% blend, since it becomes very difficult to distinguish between individual droplets which have become very small due to breakup. From the results concerning droplet size and blend morphology we can conclude that, under the conditions investigated in this work, the steady-state morphology during flow in a more confined environment is quite different from that in a bulk environment. A one-layer structure already occurs when the droplets still have enough space to form more than one layer. The so-called pearl necklace patterns are observed as a stationary morphology with constant interparticle distance z and internecklace spacing y. However, the mean droplet size is still governed by the relations that describe the structure development in nonconfined situations. Droplet Deformation. The steady-state droplet deformation is well understood in bulk flow, at least for blends consisting of Newtonian components. There, the deformation only depends on p and Ca. Pathak et al.22 also studied the deformation of confined drops in a 9.7% PDMS/PIB blend and compared the experimental results with Taylor’s theory.1 They reported serious deviations from theory for large drops with Ca > Cacrit. Here, the deformation of single droplets under confinement is investigated for varying blend concentrations and shear rates. The results are compared with the phenomenological MM model13 for droplet deformation during simple shear flow. In the model, the aspect ratio rp of the droplets is given by
rp )
f2(Ca) + [f12 + (Ca)2]1/2 - [f12 + (Ca)2 - f22(Ca)2]1/2 f2(Ca) - [f12 + (Ca)2]1/2 + [f12 + (Ca)2 - f22(Ca)2]1/2 (8)
Figure 6. Droplet aspect ratio rp versus capillary number Ca at shear rates of (a) 0.38 s-1, (b) 0.75 s-1, and (c) 2.25 s-1 (1 wt % PIB in PDMS). SALS patterns and microscopy images at 40 µm. The full line is the prediction of the MM model.
with f1 given in eq 5 and f2 expressed as
f2 )
3(Ca)2 5 + 2p + 3 2 + 6(Ca)2
(9)
In Figure 6, droplet deformations of a 1% blend at shear rates of 0.38, 0.75, and 2.25 s-1 are shown. L and B are calculated from the microscopy images using the assumption of the MM model for the orientation angle and conservation of volume. The insets in the graphs show the corresponding SALS patterns and micrographs at 40 µm. The aspect ratio rp is expressed as a function of Ca, which in the case of a constant shear rate represents the dimensionless droplet size. The full lines are the predictions of the MM model. For the 1% blend, the results at gap spacings of 40 and 100 µm are in good agreement with each other for all the shear rates applied. Both the experimental results and the model predictions correspond well in the low capillary number region, where Ca is smaller than the critical value of 0.46. Therefore, the simple phenomenological model can still be used to express droplet deformation in confined situations as long as Ca < Cacrit. Surprisingly, at a shear rate of 2.25 s-1, droplets stay stable under flow up to capillary numbers of 2 times the critical value, and at gap spacings of both 100 and 40 µm. This feature has also been observed by Pathak et al.22 for a 9.7% PDMS in
2278 Langmuir, Vol. 22, No. 5, 2006
Figure 7. Droplet aspect ratio rp versus capillary number Ca at a shear rate of 0.38 s-1 (5 wt % PIB in PDMS). SALS pattern and microscopy image at 40 µm. The full line is the prediction of the MM model.
PIB blend at a gap spacing of 36 µm. In his research, more deformed droplets were seen, as is clearly not the case here; the deformation under the present circumstances is less than predicted by the MM model. The discrepancy, however, occurs in both the confined and the nonconfined situations, and the experimental results at 2.25 s-1 are still in agreement with those at lower shear rates. At the lowest shear rate (0.38 s-1), giving rise to low capillary numbers, the SALS pattern is almost isotropic during flow, indicating a small deviation from spherical shape, which is confirmed by the micrograph. At 0.75 s-1 (Figure 6b), which corresponds with intermediate capillary numbers, some secondary streaks are clearly present, indicating the arrangement of droplets on pearl necklaces. The micrograph clearly confirms the alignment of the droplets, but this does not seem to influence the deformation of the droplets. For all the experiments performed with a gap of 40 µm for the 1% blend, 2R/d was below 0.3 for all the droplets under investigation. At a shear rate of 2.25 s-1, the droplets flow no longer in a single layer between the plates and the ordered structure has disappeared, since the droplets have become smaller due to breakup. This is confirmed by the SALS pattern at 2.25 s-1. The elliptical shape of the scattered pattern represents the deformation of single droplets, and the lack of secondary streaks is in agreement with the reestablished disorder. Similar experiments have been performed on the 5% blend. At a shear rate of 0.38 s-1, a remarkable difference between bulk behavior and confinement is noticed (Figure 7): at a gap of 100 µm, still good agreement between experimental droplet deformation and the MM model is present, but at 40 µm, the droplets clearly deform more than predicted. A possible explanation could be that droplets under confinement are more oriented in the flow direction than predicted by the theory of Maffettone and Minale. This would result in an overestimation of the aspect ratio. Calculating the droplet deformation assuming θ ) 0° still results in larger aspect ratios than the predictions of the MM model, especially for the larger Ca. Pathak et al.22 also observed deviations from the models, but for capillary numbers above Cacrit. Here, however, the capillary numbers in Figure 7 are all below the critical value of 0.46 and the steady-state droplet size is still well predicted by bulk theories as demonstrated in Figure 5, but nevertheless, deviations are seen. For the other shear rates (0.75 and 1.5 s-1), leading to droplet capillary numbers in the neighborhood of the critical value of 0.46, again good agreement between the results at gap spacings of 40 and 100 µm is seen. As can be seen in the micrograph and the SALS pattern in Figure 7, at a 40 µm gap, droplets are organized on pearl necklaces. Apparently, the width W in the vorticity direction of droplets
Vananroye et al.
Figure 8. Comparison of the droplet width as a function of the initial droplet radius at a shear rate of 0.38 s-1 (5 wt % PIB in PDMS).
Figure 9. Steady-state morphology at 40 µm (γ˘ ) 0.75 s-1). Inset: SALS pattern (10 wt % PIB in PDMS).
organized on a pearl necklace seems constant, irrespective of the size of the droplet. This is further investigated for the 5% blend. In Figure 8, the width W of the droplets during steady-state shearing at 0.38 s-1 is depicted as a function of the droplet radius R for gaps of 40 and 100 µm. The prediction of the MM model13 for the width W in simple shear flow has been added for comparison; it is given by
W ) 2R
[
f12 + (Ca)2 - f22(Ca)2
(f12 + (Ca)2)1/3(f12 + (Ca)2 - f22(Ca)2)2/3
]
1/2
(10)
in which f1 and f2 are as defined in eqs 5 and 9. It can be seen in Figure 8 that the width of the nonconfined droplets, at a gap spacing of 100 µm, is in good agreement with the prediction of the MM model. For the confined situation, however, deviations start to appear when the confined droplets reach a certain size. For 2R/d > 0.4, the droplets’ width W remains constant. As a consequence, the larger droplets are extended more in the velocity direction. Therefore, the droplets are more elongated than under bulk conditions, which is in agreement with the results from Figure 7. This constant width phenomenon can be used as a diagnostic tool to make the transition toward confined behavior. The constant width for droplets organized in pearl necklace structures was not reported by Pathak et al.21 Also a 10% blend has been subjected to simple shear flows to study the morphology development. At this concentration, the formation of superstructures can be seen at certain shear rates. A micrograph of this is shown in Figure 9, where the blend is sheared (0.75 s-1) at a gap spacing of 40 µm. The inset shows the corresponding SALS pattern. The SALS image is very anisotropic considering the low shear rate, indicating some very extended fibrils as is confirmed microscopically. Migler18 described the formation of strings as a four-step process: droplet growth, organization of the droplets on pearl necklaces,
Structure DeVelopment in Confined Polymer Blends
Langmuir, Vol. 22, No. 5, 2006 2279
coalescence of the droplets to form strings, and coalescence of the strings and ribbons. The first three steps have also been observed in our experiments on a 10% blend, though no ribbonlike structures (where W > B) were observed. From the micrograph it is clear that all strings have the same width, as was also seen with the pearl necklace structures, and as was also reported by Pathak et al.22 for strings. At a shear rate of 1.5 s-1, a bulklike morphology is reestablished for the 10% blend. It can be concluded that concentration even has a larger effect on the blend morphology and droplet deformation under confined flow than it has in bulk flow. Droplet Relaxation. In this section, the shape relaxation of single droplets after shearing is studied. After the steady state is reached at each shear rate ((5000 strain units), the shearing is temporarily stopped, allowing the droplets to retract back to their spherical shape. Several researchers used the deformed drop retraction method (DDRM) to obtain information about the interfacial tension.2,33-35 Often, the assumption is made that the two minor axes B and W are equal to each other, leading to a circular cross-section of the droplet.33,34 Because droplets under confinement are possibly flattened, asymmetry of the axes B and W could be present. Therefore, the deformation model with equal minor axes is not appropriate here. Instead, the retraction method of Mo et al.35 will be applied. In this method the droplet shape is described by a symmetric, positive-definite, second-rank tensor S, the eigenvalues of which, λ1, λ2, and λ3, represent the squared semiaxes of the ellipsoid. Hence, to define the droplet deformation during relaxation, the shape parameter (λ1 - λ2) is used35
λ 1 - λ2 )
L2 - B2 4
(11)
With this parameter, the droplet retraction can be expressed without any assumption about the shape of the drop:
[
40(p + 1) R t λ1 - λ2 ) (λ1 - λ2)0 exp ηmR0 (2p + 3)(19p + 16)
[ ]
) (λ1 - λ2)0 exp -
t τd
]
(12)
Here, t, R0, and (λ1 - λ2)0 are the time, the radius of the relaxed droplet, and the shape parameter just before the shearing is interrupted; τd represents a characteristic droplet shape relaxation time. In Figure 10, the shape relaxation of droplets with various diameters is investigated after shearing at 0.38 s-1 for a 5% blend with a gap spacing of 40 µm. As shown before (Figure 4b), a specific order is seen under these conditions as a consequence of confinement. The results in Figure 10a do not lead to a single relaxation curve. The relaxation becomes slower when the droplet size increases, which could be an effect of the confined environment. It should however be noted that similar observations were made in the case of droplet relaxation in bulk blends by Vinckier et al.36 This behavior was related to the increased aspect (33) Luciani, A.; Champagne, M. F.; Utracki, L. A. J. Polym. Sci., Part B: Polym. Phys. 1997, 35, 1393-1403. (34) Guido, S.; Villone, M. J. Colloid Interface Sci. 1999, 209, 247-250. (35) Mo, H.; Zhou, C.; Yu, W. J. Non-Newtonian Fluid Mech. 2000, 91, 221-232. (36) Vinckier, I.; Moldenaers, P.; Mewis, J. Rheol. Acta 1999, 38, 65-72.
Figure 10. (a) Single droplet relaxation after steady shearing at 0.38 s-1 (5 wt % PIB in PDMS at a gap spacing of 40 µm) and (b) data from (a) corrected for droplet deformation.
ratio for the larger droplets. By simply scaling the droplet shape relaxation time with the aspect ratio rp to the power 2/3
τ ) τdrp2/3
(13)
a master curve for all the data could be obtained. The same scaling applies to our experiments, as illustrated in Figure 6b. Hence, no effect of the confining walls can be detected under the present conditions. As a reference, relaxation of a droplet with a radius of 8.7 µm in a 100 µm gap has been investigated (full line in Figure 10b). No significant difference can be observed with the relaxation of droplets at 40 µm. From the results of droplet relaxation with a gap of 100 µm, an estimate for the interfacial tension R can be made by fitting the data with an exponential decay function. The resulting value of 2.9 mN/m is in good agreement with the values reported by Sigillo et al.,25 which are between 2.1 and 3.4 mN/m. From the study of droplet retraction, it can be concluded that, for the present conditions, confined droplet relaxation is still governed by the relations for nonconfined situations.
Conclusions The structural development of immiscible polymer blends in confined geometries was investigated during simple shear flow. It was shown that the mean droplet size during shearing is still governed by the relations that describe nonconfined situations. Nevertheless, the blend morphologies of 1% and 5% blends under confined and bulk conditions are quite different. Droplets organize in a single layer between the confining walls, even when the ratio of the droplet size to the gap is 0.25. On the basis of the results on 1%, 5%, and 10% blends, it can be concluded that
2280 Langmuir, Vol. 22, No. 5, 2006
concentration has a large effect on the blend morphology under confinement. At a 5% mass fraction of the dispersed phase with a 40 µm gap, the pearl necklace morphology is stable, and no further transition to strings and ribbons will occur. The droplets arranged in a pearl necklace all seem to keep the same width above some critical droplet diameter over gap size during shear (∼0.4). This feature can be used as a diagnostic tool to make the transition toward confined behavior. Experiments performed on a 10% blend show that strings, all having the same width, are stable under flow, but no ribbons are seen. The strings are formed by the coalescence of subsequent droplets on a pearl necklace. The width of the droplets on a necklace remains constant at a
Vananroye et al.
certain degree of confinement, giving rise to more extended structures and eventually superstrings. From the study of droplet retraction, we can conclude that droplet relaxation under geometrical confinement is still governed by the relations in nonconfined situations, as long as the effect of the aspect ratio is taken into account. Acknowledgment. FWO Vlaanderen (Project G.0523.04) and Onderzoeksfonds K.U.Leuven (Grant GOA 03/06) are gratefully acknowledged for financial support. LA0527893
