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Small-angle light scattering study of droplet break-up in emulsions and polymer blends
PII:
Chemical Engineering Science, Vol. 53, No. 12, pp. 2231—2239, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0009–2509(98)00057–8 0009—2509/98 $19.00#0.00
Small-angle light scattering study of droplet break-up in emulsions and polymer blends J. Mewis, H. Yang, P. Van Puyvelde, P. Moldenaers and L. M. Walker*s Department of Chemical Engineering, Katholieke Universiteit Leuven, B-3001 Leuven Belgium (Received 3 August 1997; accepted 19 January 1998) Abstract—The potential of using small—angle light scattering (SALS) to probe morphological changes induced by flow in immiscible polymer blends is investigated. Well—defined flow histories are shown to result in SALS patterns that are characteristic for the morphology involved. The pertinent structural change caused by either suddenly applying flow or drastically increasing the shear rate is the stretching of inclusions into long filaments, which subsequently break up by Rayleigh instabilities. Scattering models are developed to calculate the SALS patterns resulting from a filament with a sinusoidally disturbed surface and from a series of aligned spheres. These models capture the main features of the measured SALS patterns and are used to extract quantitative morphological information of the system. This is demonstrated by comparing calculated and measured results for droplet and filament size. In this manner an in situ, time-resolved technique becomes available to follow flow-induced structural changes such as those occurring during processing of blends. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Immiscible polymer blends; emulsions; small-angle light scattering; rheo-optics, dispersion.
1. INTRODUCTION
The final mechanical properties of two-phase polymer blends depend strongly on the morphology imparted during processing. Understanding the connection between applied flow fields and morphology is vital to the intelligent processing and control of blend properties. The development of techniques capable of providing in situ morphological information is a key step in this understanding. The purpose of this work is to demonstrate that rheo—optics, specifically small-angle light scattering (SALS), is a viable technique for following changes in the morphology of two-phase polymer blends and other liquid—liquid systems. A considerable understanding exists about the influence of flow fields on single droplets immersed in a second phase when both fluids are Newtonian. Experiments were originally performed by Taylor (1932, 1934) on single drops in simple shear and planar extensional flows. That work showed that a droplet, which has been extended to a filament by means of
* Present address: Department of Chemical Engineering (Colloids, Polymers and Surfaces Program), Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A. s Corresponding author.
flow, breaks up into a series of equally spaced drops when the flow is stopped. A theoretical study of this phenomenon by Tomotika (1935) demonstrated that a liquid filament in a quiescent viscous matrix disintegrates due to the action of Rayleigh instabilities at the surface. Tomotika also showed that the growth rate and dominant wavelength of this instability are a function only of the viscosity ratio between the two fluids, the interfacial tension and the initial radial dimension of the filament. More recently, a number of comprehensive experimental studies on single drops in controlled flow fields (Grace, 1982; Torza et al., 1972; Mikami et al., 1975; Bentley and Leal, 1986; Stone et al., 1986; Stone and Leal, 1989; Elemans et al., 1993; Janssen and Meijer, 1993) have led to an understanding of the influence of viscosity ratio, interfacial tension and flow type on droplet deformation and break-up. Much of this work is summarized in two review articles (Rallison, 1984; Stone, 1994). All of these studies have been performed by using microscopy to probe the evolution of the shape of a single droplet during various flow and relaxation conditions. Although considerable fundamental understanding of morphological changes under flow has resulted, the extension to probing real polymer blend systems (rather than single drops) has not been accomplished. Hence, it has not been possible to
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relate morphological changes to macroscopic flow properties. Small-angle light scattering has been used to measure droplet size in flowing two-phase systems (Sondergaard et al., 1992, 1996). In principle, the shape of the inclusions may also be derived from the SALS patterns. Unfortunately, two-phase blends under flow are polydisperse and the influence of polydispersity and shape on the SALS patterns are not easily separable, leading to a level of ambiguity in the results and limiting their general acceptance. The potential of linear conservative dichroism as a rheo-optical probe during flow has been demonstrated recently (Yang et al., 1998). The response of the dichroism could be associated with the existence of different regions of morphological behaviour. The time scales for structural transitions, i.e. droplet break-up or coalescence, have been evaluated in this manner. However, a direct link between the dichroism signal and morphology could only be made in limited cases due to the complexity of the structure and the expressions for dichroism. In this work, the possibility of deriving morphological information from SALS measurements on well-defined transient flow experiments is investigated. 2. MODELLING OF THE SCATTERING PATTERNS
When a high enough shear rate is applied to an immiscible polymer blend the droplets of the disperse phase are extended into long, slender filaments. After removal of the external flow field, surface instabilities induce break-up of the filaments (Tomotika, 1935). The present work focuses on this break-up phenomenon, so it becomes essential to be able to analyse the corresponding SALS patterns. The instabilities develop as wavy disturbances on the surface that have, at any one time, a specific wavelength and amplitude. The amplitude grows until it reaches a point at which the filament fragments into a series of aligned spherical droplets. This succession is shown schematically in Fig. 1. The physical parameters for the smooth cylinder (R , l), wavy cylinder (RM , l, j , a) and a series of 0 R
Fig. 1. Schematic representation of the three stages of break-up of an extended filament. Parameters are defined in the text.
spheres (R , s) need to be incorporated into scatter$301 ing models so that the influence of each on the intensity and direction of scattered light can be determined. Expressions for the SALS patterns are derived from Fraunhofer diffraction theory (van de Hulst, 1957). The experimental results (see next section) indicate that the conditions for this theory are satisfied although sometimes at the limit for very thin filaments. 2.1. Filaments with sinusoidal surfaces The diffraction function D(h, /) which defines the amplitude of the light diffracted by a single object onto a distant plane (see Fig. 2) is defined by (van de Hulst, 1957): 1 D(h, /)" G
PP
exp [!ik(x cos / G #y sin /) sin h] dxdy
(1)
where x and y are the coordinates of an arbitrary point within the object. The incident light is parallel to the z axis and h, / are polar and azimuthal scattering angles. The area G":: dx dy is the shadow of the G scattering object projected onto the plane P. The wave number of the light in a matrix with refractive index n is k"2nn /j with j the wavelength of the light in m m vacuum. The intensity of the scattered light, which is the measured quantity in light scattering experiments is given by
A B
G2 D D(h, /)2D, I"I 0 j2r2
(2)
where I is the incident intensity (Fig. 2). 0
Fig. 2. Geometry of scattering experiment. The coordinate frame is defined with respect to the initial filament aligned along the x axis. The incident light travels in the z direction. The scattering angles (h, /) are shown with respect to the scattering plane P.
Droplet break-up in emulsions and polymer blends
For spheres and cylinders, the form of the diffraction function D(h, /) is documented (van de Hulst, 1957). For objects with periodic surfaces, the diffraction pattern for a screw-like object has been calculated and compared with experimental results (Sethuraman, 1986). Diffraction patterns from infinite sinusoidal surfaces have been derived with the general Beckmann formulation (Fan and Huynh, 1993). The diffraction function for a cylinder with a sinusoidally varying surface has not been reported in the literature and is derived here. To describe the surface of a sinusoidally disturbed cylinder, the shape evolution predicted by Tomotika (1935) for a filament disturbed by Rayleigh instabilities is used. This surface is represented by (Fig. 1): R(x)"RM #a sin (2nx/j ) R
(3)
where j and a are the wavelength and amplitude of R the disturbance. The average radius RM "(R2! 0 a2/2)0.5 is related to both the radius of the undisturbed cylinder R and the disturbance amplitude a. 0 Note that the maximum feasible value for a is (2/3)0.5 R +0.82R (Janssen, 1993); amplitudes greater than 0 0 this value correspond to a fragmented filament and are not physically meaningful. Assuming that the incident light is perpendicular to the revolution axis of the filament, as shown in Fig. 1, the area factor is calculated to be
AB
2aj nl G"2RM l! R sin . n j R
(4)
G"2RM l.
(5)
Combining Eqs (1), (3) and (5) and defining c"k sin / sin h and d"k cos / sin h, the diffraction function is re-expressed as
P
A
A BB
1 l@2 2nx D(h, /)" sin cRM !ac cos cRM l j ~l@2 R
e~idx dx (6)
and rewritten as 1 D(h, /)" [sin(cRM ) A!cos(cRM ) B], cRM
(7)
with
P P
A BB A BB
Here A and B contain all the dependence on the disturbance wavelength. The integrals in A and B may be expressed with a Neumann expansion series (Korn and Korn, 1961):
AB
dl = A"J (ac) sinc # + (!1)i J (ac) 0 2i 2 i/1 4in l 4in l ] sinc #d #sinc !d j 2 j 2 R R
G AA
1 l@2 2nx A" cos (ac cos e~*$x dx l j ~l@2 R 1 l@2 2nx B" sin(ac cos e~*$x dx. l j ~l@2 R
(8)
B B AA G AA B BH
B BH BB
= 2(2i#1)n l B" + (!1)iJ (ac) sinc #d 2i`1 j 2 R i/0 2(2i#1)n l #sinc !d j 2 R
AA
(9)
where sinc (x) is the function sin (x)/x. Using these expressions, the intensity of scattered light on the scattering plane is determined. When the amplitude of the disturbance a is zero, J (0)"1, and the higher 0 orders of the Bessel functions are zero, so that the Fraunhofer diffraction function for a cylindrical object is recovered (van de Hulst, 1957). The expression for the scattering angle dependency of the light intensity along the /"0 axis at each moment is given by (d"k sin h):
A AB AA B BDB
I(h, /"0)J D D(h, /"0) D2" sinc
C AA
] sinc
The filament is assumed to be long and the wavelength of the disturbance much shorter than the total filament length (l
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BB
2n l #d #sinc j 2 R
a ! 2RM 2
dl
2n l !d j 2 R
2 . (10)
This will be used in Section 2.3 to demonstrate the effect of the model parameters on the measured SALS patterns. 2.2. Aligned spherical droplets As noted earlier, the amplitude of the surface disturbance is bounded by a+0.82R . When the amplitude o of the disturbance reaches this limit, the morphology changes to one of a string of independent spheres. In reality, small satellite drops are observed between larger spheres (Elemans, 1989) but, as the size of structures probed via SALS is limited, we assume that all the scattering arises from the larger spheres. The diffraction function D(h, /) for a single sphere is known (van de Hulst, 1957). However, theoretical treatment of the scattering of an array of equally spaced spheres may not be tackled with the diffraction function approach that considers only scattering from a single object. Instead, an expression for a string of spheres is calculated by starting from the Fresnel extension of the Huygen principle. The electric field of the scattered light at point P in plane incident light (see Fig. 2) is expressed as (van de Hulst, 1957)
P
i
E(h, /)"
G1`G2`G3`2
e~*kr E dG 0 j r
(11)
J. Mewis et al.
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with E the incident electric field. The integration 0 is performed over the projected areas of all the objects, in this case the spheres. The distance to the scattering plane r is assumed to be much larger than the dimension of any sphere or the distance between the spheres. The electric field generated by equally spaced spheres is given by i
E(h, /)" E e~*kr0 + jr 0 j
PP
e~*k(x#04(`y4*/()4*/h dx dy Gj
(12)
where the integration extends over j spheres. The summation simply expresses the superposition of the coherent diffraction patterns of the spheres. As the spheres are identical, eq. (12) may be simplified further by coordination transformation: n i E(h, /)" E e~*kr0 + e~*k(2R$301`s)j#04(4*/h 0 jr j/1
PP
]
e~*k(g#04(`m4*/()4*/h dg dm (13) G with R the radius of an individual sphere and s $301 the spacing between the spheres. The integration is now over only a single sphere of projected area G"nR2 . The integral in eq. (13) corresponds to the $301 diffraction function from a single sphere which is known (van de Hulst, 1957): J (kR sin h) D (h, /)"2 1 $301 . s kR sin h $301
(14)
By choosing the midpoint of the central sphere as the symmetry point, eq. (13) becomes
C
iGE (n~1)@2 0 e~*kr0 1#2 + cos ( jk(2R E(h, /)" $301 jr j/1
D
#s) cos / sin h) D (h, /). s
(15)
Consequently, with I "D E D2, the intensity of the 0 0 scattered light by the array of spheres is given by
C
(n~1)@2 G2I 0 1#2 + cos ( jk(2R I(h, /)" $301 j2r2 j/1
D
#s) cos / sin h)
2
D D (h, /) D2. s
(16)
2.3. Influence of morphological parameters Both of the scattering models must be solved by evaluating a summation. The numerical software package Mathematica is used to generate intensity distributions for comparison with experimental SALS patterns. Values of I(h, /) are calculated for polar angles between 0° and 11°. For the filaments with sinusoidal surfaces, the solution requires a summation over a series of Bessel functions. The general shape of the intensity distribution is obtained with two terms [i"2 in eq. (9)]. Higher order terms act only to minimize the lower amplitude oscillations that are caused by truncating the series. Similarly, the
Fig. 3. Iso intensity contour plots demonstrating the influence of parameters in Fig. 1 on the calculated two—dimensional SALS patterns. Varying amplitude of the disturbance at a fixed wavelength (a, b, c), varying wavelength at a fixed amplitude (c, d) and the resulting pattern for an array of spheres (e).
Droplet break-up in emulsions and polymer blends
smoothness of the aligned sphere model [eq. (16)] is determined by the number of spheres. Figure 3 shows iso-intensity plots of typical scattering patterns calculated with the diffraction models for filaments with sinusoidal surfaces (a, b, c, d) and aligned spherical droplets (e). The first three plots (a, b, c) demonstrate the effect of varying the amplitude of the disturbance at constant wavelength, while a comparison of plots (c) and (d) illustrates the change in scattering pattern with different wavelengths at a constant amplitude. For a"0 the scattering pattern is that of a cylinder (R "0.5 km), appearing as a streak 0 that is oriented perpendicular to the orientation direction of the filament. When a becomes non-zero, secondary streaks appear parallel to the primary streak. The intensity of these streaks depends on the value of a while their position is determined by the value of the wavelength j . As mentioned before, the amplitude R growth of the disturbance is bounded by a+0.82R 0 above which value the diffraction model for aligned spheres must be used. The smooth transition between the two models is demonstrated by the last two plots in Fig. 3. Plot (d) corresponds to a filament near breaking, with a"0.8R , while plot (e) shows the 0 predicted scattering for a series of aligned spheres. The influence of the different physical parameters is observed more easily by comparing the intensity measured along the /"0 axis, i.e. the intensity parallel to the orientation of the original filament. The amplitude of the disturbance is related directly to the amplitude of the intensity in the secondary streak [see Fig. 4(a)], while Fig. 4(b) demonstrates that varying the wavelength at a fixed value of a influences the position h of the secondary streak. As the wavelength is inm creased, the streak moves to smaller scattering angles through the relation kj sin h "2n [see eq. (10)]. R m The ability to clearly separate the influence of these two physical parameters on the features of the SALS patterns makes it possible to follow the development of the surface disturbance via SALS. If the forma-
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tion of the satellite droplets is neglected, the spacing between the aligned spherical drops is fixed by the wavelength of the disturbance through j " R 2R #s. The intensity of the secondary streak is $301 then controlled by the relative magnitude of R with respect to s. Since the formation of the $301 and satellite droplets is neglected, the values of R $301 s of the spheres can be deduced from R and the 0 wavelength of the disturbance. Therefore, the position of the secondary streaks after the filament breaks may be measured to investigate the final state of the morphology. The relationship kj sin h "2n provides a quick R m method for estimating the magnitude of the wavelength of the disturbance in real units. Unfortunately, the wave number includes the refractive index, which is often difficult to determine for a concentrated blend system. However, it is constant within a single blend, so the relationship is still quite powerful when comparing the results of different flow histories. Combining the two models discussed above, predictions can be made of the SALS patterns resulting from the development of Rayleigh instabilities on long filaments. The Tomotika analysis can be used to study break-up after cessation of shear flow, at least in the case of purely Newtonian fluids. The amplitude is expected to grow at constant wavelength and consequently the secondary streaks should display an increasing light intensity but should not change their position. 3. EXPERIMENTAL
3.1. Materials The materials used in the present investigation are polydimethylsiloxane (PDMS) from RhoˆnePoulenc (Rhodorsil 47V200.000) and polyisobutene (PIB) from Exxon Chemical (Parapol 1300). The steady state rheological properties of both materials have been measured with a Rheometrics Mechanical
Fig. 4. Calculated SALS intensity along the /"0 axis for the parameters used in Fig. 3. Part (a) is for fixed j "6 km with amplitude varying from a smooth cylinder to near breaking. Part (b) is for fixed amplitude R and changing j as well as a series of aligned spheres R "0.91 km and s"2.18 km. R $301
J. Mewis et al.
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Table 1. Material properties of the blend components
Polymer
Density (kg/m3)
n
g o (Pa s)
( 1,0 (Pa s2)
PDMS PIB-1300
975 895
1.406 1.499
200 90
10 0.06
Note: All values reported at 23°C.
Spectrometer RMS800 using cone and plate geometry (radius 12.5 mm, cone angle 0.1 rads) at 23.0°C (Table 1). The maximum shear rate in these experiments is limited by shear fracture, especially for the more elastic PDMS sample. The viscosities of the two materials are constant over the accessible range of shear rates. The PDMS is about twice as viscous as the PIB. Both polymers display some degree of elasticity and behave approximately as second-order fluids, characterized here by the first normal stress coefficient (( ). This 1,0 factor is about two decades higher for PDMS than for PIB. Blends consisting of mixtures of 1% by weight of PDMS in PIB are prepared using a turbine mixer with a six flat-blade disk. The freshly mixed blend is placed in a vacuum oven at room temperature to eliminate air bubbles. The interfacial tension ! between the two phases in the PDMS/PIB-1300 system is 3.1 mN/m (Sigillo et al., 1997) and the viscosity ratio p+2 at 23°C. 3.2. Experimental procedure For the system under investigation the droplet size is governed by the shear history, hence flow can be used to generate a well—defined initial condition for transient experiments. In the present experiments, a preshear at a shear rate c5"2 1/s is applied until the scattering pattern does not change with further shearing, in this case, after approximately 4000 strain units. The shear rate is then increased suddenly to 30 1/s for a short time. The preshear at 2 1/s acts to break up large droplets, but is not strong enough to greatly deform the smaller droplets. Based on previous work (Vinckier et al., 1996), it is known that the steady-state morphology is made up of slightly deformed droplets of radius R corresponding to those defined by the critical value of capillary number, Ca,g c5R/!, with m g the matrix viscosity and c5 the shear rate. The size of m the droplets is extracted from SALS measurements providing a radius of 2.5$0.5 km (Yang et al., 1997). After increasing the shear rate suddenly, the droplets deform affinely (Elemans, 1989) to high levels of l/R o so that, when the flow is stopped the morphology consists of extended fibrils of PDMS in a quiescent matrix of PIB. 3.3. Small-angle light scattering The flow small-angle light scattering apparatus (Fig. 5) used in this work is a modification of a Rheometrics Optical Analyser (ROA). It has a polarized 10 mW He—Ne laser light source (j"633 nm).
Fig. 5. Modified Rheometrics ROA used to capture SALS patterns of the blend system.
In these experiments, the polarization direction is continuously rotated by a half-wave plate (rotating at 2000 rps) before it reaches the sample cell. The sample is sheared in a parallel plate shear cell consisting of two optical grade glass plates. The upper platen of the cell is driven by a stepper motor with an angular precision of 25,000 steps per revolution. The beam passes through the sample along the velocity gradient direction at a distance of 0.53 cm from the centre of rotation. A sample thickness of 200 km is maintained to limit multiple scattering in these fairly turbid samples. The light scattered by the samples is visualized on a semi-transparent plane parallel to the velocity—vorticity plane, positioned at 4.95 cm below the cell. The maximum accessible scattering angle is about 15°. A circular beam stop is used to block the main beam and to avoid saturation of the camera. A color CCD camera (Ikegami model ICD-810P) is mounted in the optical track to record the transient scattering patterns. The patterns are either captured on video tape or directly stored on an IBM-PC, which is equipped with a frame grabber (Data Translations DT3851), for further analysis of the scattering patterns.
4. RESULTS AND DISCUSSION
Figure 6 shows the evolution of experimental SALS patterns for 1% PDMS in PIB during the shear and relaxation experiments. The sample is presheared at 2 1/s and then the shear rate is increased suddenly to 30 1/s for 4 s to create highly elongated filaments that are oriented close to the shear direction. The flow is
Droplet break-up in emulsions and polymer blends
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Fig. 6. Measured SALS patterns for 1% PDMS/PIB after a two-step shearing (preshear at c5"2 1/s, step up to 30 1/s for 4 s, then relaxation).
then stopped and the interface created by the extension of the inclusions is allowed to relax by way of surface instabilities. In the SALS patterns, a diffuse ellipsoidal scattering pattern is observed during the preshear at 2 1/s, with the major axis perpendicular to the flow direction. It reflects the presence of slightly deformed droplets. Upon increasing the shear rate suddenly to 30 1/s, the pattern distorts rapidly into a bright streak perpendicular to the flow direction, indicating that the high shear rate effectively stretches the droplets in the flow direction. Once the flow is stopped, streaks grow parallel to the primary streak in the SALS pattern. These secondary streaks increase in intensity during several seconds and then remain unchanged even for hours. The development of the SALS patterns after flow demonstrates that droplet retraction does not dominate the relaxation mechanism. Retraction would cause the scattering pattern to become circular rather than showing multiple streaks after long periods at rest. In Fig. 7 the evolution of the intensity along the /"0 axis is taken from the two-dimensional patterns in Fig. 6. From the previous section it is known that the position h of the streak is related to the m wavelength of the disturbance which causes the filaments to break up. At long times the position of h reflects the spacing of the resulting aligned spherim cal droplets. The variation of intensity in the streak is indicative of the growth of the amplitude of the disturbance and the formation of aligned spheres. Because the secondary streaks are initially weak in intensity, it
is impossible to determine if the position of the streaks varies immediately after stopping the flow. In this experiment, the appearance of the streaks, observed as peaks in Fig. 7, can be seen to clearly appear at 0.5 s after the flow is stopped. The diffraction model for a filament with an oscillating surface allows for an independent measure of j and the growth rate of the disturbance a(t) . From R the model, a relation between the position h of the m streak and the wavelength is obtained: h "arm csin(2n/kj ). In the experimental intensity profile R (Fig. 7), sharp peaks are not evident. This is due to a number of factors including background scatter and the fact that there will be a distribution of filament diameters resulting from a polydisperse distribution of initial droplet sizes. Although it is possible to account for background scatter and develop a model to account for polydispersity it should be noted that the weak peak in Fig. 7 may be picked out with far more accuracy than trying to extract size information from a monotonic intensity distribution, such as the diffraction pattern resulting from spheres, ellipsoids or cylinders. Therefore, in this work we consider only the peak position and relate it to an average filament diameter. For this system, the secondary peak in I(h, /"0) is observed at 6.0$0.4o, hence the value of j of the disturbance for these conditions corresponds R to 4.0$0.3 km. For a given dilute dispersion, consisting of a Newtonian fluid in another Newtonian fluid, Tomotika (1935) calculated the wavelength of the dominant disturbance which has the fastest growth
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J. Mewis et al.
Fig. 7. Intensity profiles along /"0 (from the SALS patterns in Fig. 6) during break-up of the fibrils. The maximum at h corresponds to the position of the secondary streaks. m
rate and eventually causes the filaments to break. The relationship between the dominant wavelength and initial filament radius is determined by the viscosity ratio of the dispersion (Tomotika, 1935; Mikami et al., 1975; Janssen, 1993). In the case of p"2, one finds R "0.52j /(2n) . This may be combined with the 0 R,m measured value of j "4.0 km to provide an estiR,m mate of the initial radius of the filament; i.e., R "0.3 km. A simple mass balance provides the size o and spacing of the resulting string of independent droplets after break-up. For the given flow history, the final morphology is a series of drops of R "0.7 km evenly spaced at s"2.6 km apart. $301 This is a direct measure of the effectiveness of droplet dispersion by shear and Rayleigh instabilities. A single large droplet created at a rate of 2 1/s is broken up into a number on the order of 100 smaller droplets. Inclusions of an initial radius of several micrometers are dispersed into droplets of radius 0.7 km by shearing at a high rate for only 4 s. Assuming that the initial inclusions are deformed affinely, the filament radius at cessation of flow may be estimated and compared to the one measured by SALS. Affine deformation implies that there is a direct relationship between the dimensions of the filament and the applied strain. The conditions of the flow history used here, shearing for 4 s at 30 1/s result in 120 strain units. Combined with the estimate for R derived from the SALS pattern, 2.5 km, the filament is calculated to be 612 km long. A simple volume balance of the inclusion then gives a filament radius of R "0.2 km. This value agrees reasonably well with 0 the value of 0.3 km determined from the dominant wavelength of the instability. The value determined through affine deformation strongly depends on the size of the initial inclusions which is known only approximately. The value of R determined by ap0 plying the model to the SALS pattern comes from a direct morphological measurement and depends only
on the viscosity ratio of the two materials p, and the assumptions of the model developed by Tomotika. Therefore, the value of 0.3 km is considered more accurate. The growth rate of the instability may be determined from the temporal increase in intensity of the streak. This should follow an exponential dependence as predicted by Tomotika (1935). However, the high level of background scattering in this system, due to turbidity, makes this comparison pointless at this time. In a more ideal system extracting structural information from the intensity growth in the streaks is feasible. One should also note that eq. (13) may be solved for any shape to describe the scattering of an array of objects. For example, an analysis of the case of aligned ellipsoids should provide the scattering from the system at the breaking point. This will improve the continuity in the calculation of evolution of the intensity of the secondary streaks patterns but will not change the position of the peaks. As we analyse only the peak position in this study, this refinement to the scattering models has not been included. It should be clear that the proposed technique is more accurate than attempting to determine the radius of the filaments directly from SALS patterns at high shear rates. The size of the filaments (0.1—1.0 km) falls between scattering theories (RGD, Mie and diffraction). The scattering function from a filament (cylinder) is a decaying function of scattering angle and is difficult to fit without considerable error and ambiguity. The appearance of a streak is therefore a much more accurate way of extracting size information from small-angle light scattering. The combination of well designed flow experiments with a series of scattering models provides a powerful technique for probing the morphology. Turbidity limits the application of the technique to low volume fractions of the dispersed phase. Yet, it can be used to study the effect of parameters such as viscoelasticity or the presence of compatibilizer in dilute systems of polymers used in real blends. Experiments on model systems more conducive to scattering are possible (Van Puyvelde et al., 1998) and are being used to elucidate the role of various system parameters on filament break-up.
5. CONCLUSIONS
Combining SALS data from a well-defined shear experiment with suitable diffraction models is shown to result in a viable technique for characterizing specific morphological features of a two-phase polymer blend during flow and in particular during filament break-up. Scattering models based on Fraunhofer diffraction provide predictions of the scattering patterns resulting from extended filaments, filaments with a sinusoidal surface disturbance and a series of aligned spheres. The combination of the latter two models has led to a prediction of the SALS patterns observed during break-up due to Rayleigh instabilities. Analysis of the scattering models makes it
Droplet break-up in emulsions and polymer blends
possible to quantify separately the effects of wavelength development and disturbance growth. They can be related to measurable scattering properties. An important result is the simple relationship (h "arcsin(2n/kj )) between the position of the m R streak in the SALS pattern and the magnitude of the dominant wavelength of the surface disturbance. The materials and scattering system utilized in this work limited the investigation to low concentrations and simple verification of the technique. Use of a more accurate SALS device will allow the wavelength and growth rate of the instabilities to be probed in situ and, hence, compared to mechanical measurements of flow properties. The influence of compatabilizers, viscoelasticity and interfacial tension on the break-up phenomenon are then easily investigated. Although this technique is limited to dilute systems, the ability to probe the influence of these parameters in situ will provide vital understanding of the filament break-up process. This technique is not restricted to polymer blends, but may be applied to emulsions and other two phase systems. The present work results in a more accurate technique for determining filament radius and offers the potential for characterizing filament break-up in model blend systems. The analysis of specific features of the SALS pattern, i.e. a streak, provides a much more accurate technique for extracting length scale information from these patterns than using the monotonically decaying part of the pattern. A carefully designed experimental protocol allows morphological size information to be extracted during flow and break-up. The effect of material parameters (p, !, g ) m on the flow-induced morphology can be probed and characterized rapidly in situ with this procedure.
Acknowledgements Partial financial support for this project has been provided by the European BRITE-EURAM program (Contract No. BRE2.CT92.0213), Onderzoeksfonds K.U. Leuven and FKFO. LMW acknowledges the National Science Foundation (INT-9505545) for post-doctoral funding of this work.
REFERENCES
Bentley, B. J. and Leal, L. G. (1986) An experimental investigation of drop deformation and breakup in steady, two-dimensional linear shear flows. J. Fluid Mech. 167, 241—283. Elemans, P. H. M. (1989) Modelling of the processing of incompatible polymer blends. Ph.D. thesis, Eindhoven University of Technology. Elemans, P. H. M., Bos, H. L., Janssen, J. M. H. and Meijer, H. E. H. (1993) Transient Phenomena in Dispersive Mixing. Chem. Engng Sci. 48, 267—276. Fan, Y. Y. and Huynh, V. M. (1993) Light scattering from periodic rough cylindrical surfaces. Appl. Opt. 32, 3452—3458. Grace, H. P. (1982) Dispersion phenomena in high viscosity immiscible fluid systems and application of static mixers as dispersion devices in such systems. Chem. Engng Commun. 14, 225—277.
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Janssen, J. M. H. (1993) Dynamics of liquid/liquid mixing. Ph.D. thesis, Eindhoven University of Technology. Janssen, J. M. H. and Meijer, H. E. H. (1993) Droplet breakup mechanisms: stepwise equilibrium vs transient dispersion. J. Rheol. 37, 597—608. Korn, G. A. and Korn, T. M. (1961) Mathematical Handbook for Scientists and Engineers. McGrawHill, New York. Mikami, T., Cox, R. G. and Mason, S. G. (1975) Breakup of extending liquid threads. Int. J. Multiphase Flow 2, 113—138. Rallison, J. M. (1984) The deformation of small viscous drops and bubbles in shear flows. Ann. Rev. Fluid Mech. 16, 45—66. Sethuraman, J. (1986) Optical display of the ambiguity function and the diffraction pattern of a screw—an analogy. Am. J. Phys. 54, 450—452. Sigillo, I., di Santo, L., Guido, S. and Grizzuti, N. (1997) Comparative study of interfacial tension in a model polymer blend. Polym. Engng Sci. 37, 1540—1549. Sondergaard, K. and Lyngaae-Jorgensen, J. (1996) Coalescence in an interface modified polymer blend revealed by light scattering measurements. Polymer 37, 509—517. Sondergaard, K., Valenza, A. and Lyngaae-Jorgensen, J. (1992) Effect of a compatibilizer on the morphology of polymer blends revealed by light scattering measurements during flow. Polym. Network Blends 2, 139—165. Stone, H. A. (1994) Dynamics of drop deformation and breakup in viscous flows. Ann. Rev. Fluid Mech. 26, 65—102. Stone, H. A. and Leal, L. G. (1989) Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid. J. Fluid Mech. 198, 399—427. Stone, H. A., Bentley, B. J. and Leal, L. G. (1986) An experimental study of transient effects in the breakup of viscous drops. J. Fluid Mech. 173, 131—158. Taylor, G. I. (1932) The viscosity of a fluid containing small drops of another fluid. Proceedings of Royal Society of ¸ondon A 138, 41—48. Taylor, G. I. (1934) The formation of emulsions in definable fields of flow. Proceedings of Royal Society of ¸ondon A 146, 501—523. Tomotika, S. (1935) On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proceedings of Royal Society of ¸ondon A 150, 322—337. Torza, S., Cox, R. G. and Mason, S. G. (1972) Particle motions in sheared suspensions: XXVII. Transient and steady deformation and burst of liquid drops. J. Colloid. Interface Sci. 38, 395—411. van de Hulst, H. C. (1957) ¸ight Scattering by Small Particles. Dover, New York. Van Puyvelde, P., Yang, H., Mewis, J. and Moldenaers, P. (1998) Rheo-optical probing of relaxation phenomena in immiscible polymer blends. J. Colloid Int. Sci. 200, 86—94. Vinckier, I., Moldenaers, P. and Mewis, J. (1996) Relationship between rheology and morphology of model blends in steady shear flow. J. Rheol. 40, 613—631. Yang, H., Moldenaers, P. and Mewis, J. (1998) Rheooptical investigation of immiscible polymer blends, Polymer (in press).
Chemical Engineering Science, Vol. 53, No. 12, pp. 2231—2239, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0009–2509(98)00057–8 0009—2509/98 $19.00#0.00
Small-angle light scattering study of droplet break-up in emulsions and polymer blends J. Mewis, H. Yang, P. Van Puyvelde, P. Moldenaers and L. M. Walker*s Department of Chemical Engineering, Katholieke Universiteit Leuven, B-3001 Leuven Belgium (Received 3 August 1997; accepted 19 January 1998) Abstract—The potential of using small—angle light scattering (SALS) to probe morphological changes induced by flow in immiscible polymer blends is investigated. Well—defined flow histories are shown to result in SALS patterns that are characteristic for the morphology involved. The pertinent structural change caused by either suddenly applying flow or drastically increasing the shear rate is the stretching of inclusions into long filaments, which subsequently break up by Rayleigh instabilities. Scattering models are developed to calculate the SALS patterns resulting from a filament with a sinusoidally disturbed surface and from a series of aligned spheres. These models capture the main features of the measured SALS patterns and are used to extract quantitative morphological information of the system. This is demonstrated by comparing calculated and measured results for droplet and filament size. In this manner an in situ, time-resolved technique becomes available to follow flow-induced structural changes such as those occurring during processing of blends. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Immiscible polymer blends; emulsions; small-angle light scattering; rheo-optics, dispersion.
1. INTRODUCTION
The final mechanical properties of two-phase polymer blends depend strongly on the morphology imparted during processing. Understanding the connection between applied flow fields and morphology is vital to the intelligent processing and control of blend properties. The development of techniques capable of providing in situ morphological information is a key step in this understanding. The purpose of this work is to demonstrate that rheo—optics, specifically small-angle light scattering (SALS), is a viable technique for following changes in the morphology of two-phase polymer blends and other liquid—liquid systems. A considerable understanding exists about the influence of flow fields on single droplets immersed in a second phase when both fluids are Newtonian. Experiments were originally performed by Taylor (1932, 1934) on single drops in simple shear and planar extensional flows. That work showed that a droplet, which has been extended to a filament by means of
* Present address: Department of Chemical Engineering (Colloids, Polymers and Surfaces Program), Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A. s Corresponding author.
flow, breaks up into a series of equally spaced drops when the flow is stopped. A theoretical study of this phenomenon by Tomotika (1935) demonstrated that a liquid filament in a quiescent viscous matrix disintegrates due to the action of Rayleigh instabilities at the surface. Tomotika also showed that the growth rate and dominant wavelength of this instability are a function only of the viscosity ratio between the two fluids, the interfacial tension and the initial radial dimension of the filament. More recently, a number of comprehensive experimental studies on single drops in controlled flow fields (Grace, 1982; Torza et al., 1972; Mikami et al., 1975; Bentley and Leal, 1986; Stone et al., 1986; Stone and Leal, 1989; Elemans et al., 1993; Janssen and Meijer, 1993) have led to an understanding of the influence of viscosity ratio, interfacial tension and flow type on droplet deformation and break-up. Much of this work is summarized in two review articles (Rallison, 1984; Stone, 1994). All of these studies have been performed by using microscopy to probe the evolution of the shape of a single droplet during various flow and relaxation conditions. Although considerable fundamental understanding of morphological changes under flow has resulted, the extension to probing real polymer blend systems (rather than single drops) has not been accomplished. Hence, it has not been possible to
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relate morphological changes to macroscopic flow properties. Small-angle light scattering has been used to measure droplet size in flowing two-phase systems (Sondergaard et al., 1992, 1996). In principle, the shape of the inclusions may also be derived from the SALS patterns. Unfortunately, two-phase blends under flow are polydisperse and the influence of polydispersity and shape on the SALS patterns are not easily separable, leading to a level of ambiguity in the results and limiting their general acceptance. The potential of linear conservative dichroism as a rheo-optical probe during flow has been demonstrated recently (Yang et al., 1998). The response of the dichroism could be associated with the existence of different regions of morphological behaviour. The time scales for structural transitions, i.e. droplet break-up or coalescence, have been evaluated in this manner. However, a direct link between the dichroism signal and morphology could only be made in limited cases due to the complexity of the structure and the expressions for dichroism. In this work, the possibility of deriving morphological information from SALS measurements on well-defined transient flow experiments is investigated. 2. MODELLING OF THE SCATTERING PATTERNS
When a high enough shear rate is applied to an immiscible polymer blend the droplets of the disperse phase are extended into long, slender filaments. After removal of the external flow field, surface instabilities induce break-up of the filaments (Tomotika, 1935). The present work focuses on this break-up phenomenon, so it becomes essential to be able to analyse the corresponding SALS patterns. The instabilities develop as wavy disturbances on the surface that have, at any one time, a specific wavelength and amplitude. The amplitude grows until it reaches a point at which the filament fragments into a series of aligned spherical droplets. This succession is shown schematically in Fig. 1. The physical parameters for the smooth cylinder (R , l), wavy cylinder (RM , l, j , a) and a series of 0 R
Fig. 1. Schematic representation of the three stages of break-up of an extended filament. Parameters are defined in the text.
spheres (R , s) need to be incorporated into scatter$301 ing models so that the influence of each on the intensity and direction of scattered light can be determined. Expressions for the SALS patterns are derived from Fraunhofer diffraction theory (van de Hulst, 1957). The experimental results (see next section) indicate that the conditions for this theory are satisfied although sometimes at the limit for very thin filaments. 2.1. Filaments with sinusoidal surfaces The diffraction function D(h, /) which defines the amplitude of the light diffracted by a single object onto a distant plane (see Fig. 2) is defined by (van de Hulst, 1957): 1 D(h, /)" G
PP
exp [!ik(x cos / G #y sin /) sin h] dxdy
(1)
where x and y are the coordinates of an arbitrary point within the object. The incident light is parallel to the z axis and h, / are polar and azimuthal scattering angles. The area G":: dx dy is the shadow of the G scattering object projected onto the plane P. The wave number of the light in a matrix with refractive index n is k"2nn /j with j the wavelength of the light in m m vacuum. The intensity of the scattered light, which is the measured quantity in light scattering experiments is given by
A B
G2 D D(h, /)2D, I"I 0 j2r2
(2)
where I is the incident intensity (Fig. 2). 0
Fig. 2. Geometry of scattering experiment. The coordinate frame is defined with respect to the initial filament aligned along the x axis. The incident light travels in the z direction. The scattering angles (h, /) are shown with respect to the scattering plane P.
Droplet break-up in emulsions and polymer blends
For spheres and cylinders, the form of the diffraction function D(h, /) is documented (van de Hulst, 1957). For objects with periodic surfaces, the diffraction pattern for a screw-like object has been calculated and compared with experimental results (Sethuraman, 1986). Diffraction patterns from infinite sinusoidal surfaces have been derived with the general Beckmann formulation (Fan and Huynh, 1993). The diffraction function for a cylinder with a sinusoidally varying surface has not been reported in the literature and is derived here. To describe the surface of a sinusoidally disturbed cylinder, the shape evolution predicted by Tomotika (1935) for a filament disturbed by Rayleigh instabilities is used. This surface is represented by (Fig. 1): R(x)"RM #a sin (2nx/j ) R
(3)
where j and a are the wavelength and amplitude of R the disturbance. The average radius RM "(R2! 0 a2/2)0.5 is related to both the radius of the undisturbed cylinder R and the disturbance amplitude a. 0 Note that the maximum feasible value for a is (2/3)0.5 R +0.82R (Janssen, 1993); amplitudes greater than 0 0 this value correspond to a fragmented filament and are not physically meaningful. Assuming that the incident light is perpendicular to the revolution axis of the filament, as shown in Fig. 1, the area factor is calculated to be
AB
2aj nl G"2RM l! R sin . n j R
(4)
G"2RM l.
(5)
Combining Eqs (1), (3) and (5) and defining c"k sin / sin h and d"k cos / sin h, the diffraction function is re-expressed as
P
A
A BB
1 l@2 2nx D(h, /)" sin cRM !ac cos cRM l j ~l@2 R
e~idx dx (6)
and rewritten as 1 D(h, /)" [sin(cRM ) A!cos(cRM ) B], cRM
(7)
with
P P
A BB A BB
Here A and B contain all the dependence on the disturbance wavelength. The integrals in A and B may be expressed with a Neumann expansion series (Korn and Korn, 1961):
AB
dl = A"J (ac) sinc # + (!1)i J (ac) 0 2i 2 i/1 4in l 4in l ] sinc #d #sinc !d j 2 j 2 R R
G AA
1 l@2 2nx A" cos (ac cos e~*$x dx l j ~l@2 R 1 l@2 2nx B" sin(ac cos e~*$x dx. l j ~l@2 R
(8)
B B AA G AA B BH
B BH BB
= 2(2i#1)n l B" + (!1)iJ (ac) sinc #d 2i`1 j 2 R i/0 2(2i#1)n l #sinc !d j 2 R
AA
(9)
where sinc (x) is the function sin (x)/x. Using these expressions, the intensity of scattered light on the scattering plane is determined. When the amplitude of the disturbance a is zero, J (0)"1, and the higher 0 orders of the Bessel functions are zero, so that the Fraunhofer diffraction function for a cylindrical object is recovered (van de Hulst, 1957). The expression for the scattering angle dependency of the light intensity along the /"0 axis at each moment is given by (d"k sin h):
A AB AA B BDB
I(h, /"0)J D D(h, /"0) D2" sinc
C AA
] sinc
The filament is assumed to be long and the wavelength of the disturbance much shorter than the total filament length (l
2233
BB
2n l #d #sinc j 2 R
a ! 2RM 2
dl
2n l !d j 2 R
2 . (10)
This will be used in Section 2.3 to demonstrate the effect of the model parameters on the measured SALS patterns. 2.2. Aligned spherical droplets As noted earlier, the amplitude of the surface disturbance is bounded by a+0.82R . When the amplitude o of the disturbance reaches this limit, the morphology changes to one of a string of independent spheres. In reality, small satellite drops are observed between larger spheres (Elemans, 1989) but, as the size of structures probed via SALS is limited, we assume that all the scattering arises from the larger spheres. The diffraction function D(h, /) for a single sphere is known (van de Hulst, 1957). However, theoretical treatment of the scattering of an array of equally spaced spheres may not be tackled with the diffraction function approach that considers only scattering from a single object. Instead, an expression for a string of spheres is calculated by starting from the Fresnel extension of the Huygen principle. The electric field of the scattered light at point P in plane incident light (see Fig. 2) is expressed as (van de Hulst, 1957)
P
i
E(h, /)"
G1`G2`G3`2
e~*kr E dG 0 j r
(11)
J. Mewis et al.
2234
with E the incident electric field. The integration 0 is performed over the projected areas of all the objects, in this case the spheres. The distance to the scattering plane r is assumed to be much larger than the dimension of any sphere or the distance between the spheres. The electric field generated by equally spaced spheres is given by i
E(h, /)" E e~*kr0 + jr 0 j
PP
e~*k(x#04(`y4*/()4*/h dx dy Gj
(12)
where the integration extends over j spheres. The summation simply expresses the superposition of the coherent diffraction patterns of the spheres. As the spheres are identical, eq. (12) may be simplified further by coordination transformation: n i E(h, /)" E e~*kr0 + e~*k(2R$301`s)j#04(4*/h 0 jr j/1
PP
]
e~*k(g#04(`m4*/()4*/h dg dm (13) G with R the radius of an individual sphere and s $301 the spacing between the spheres. The integration is now over only a single sphere of projected area G"nR2 . The integral in eq. (13) corresponds to the $301 diffraction function from a single sphere which is known (van de Hulst, 1957): J (kR sin h) D (h, /)"2 1 $301 . s kR sin h $301
(14)
By choosing the midpoint of the central sphere as the symmetry point, eq. (13) becomes
C
iGE (n~1)@2 0 e~*kr0 1#2 + cos ( jk(2R E(h, /)" $301 jr j/1
D
#s) cos / sin h) D (h, /). s
(15)
Consequently, with I "D E D2, the intensity of the 0 0 scattered light by the array of spheres is given by
C
(n~1)@2 G2I 0 1#2 + cos ( jk(2R I(h, /)" $301 j2r2 j/1
D
#s) cos / sin h)
2
D D (h, /) D2. s
(16)
2.3. Influence of morphological parameters Both of the scattering models must be solved by evaluating a summation. The numerical software package Mathematica is used to generate intensity distributions for comparison with experimental SALS patterns. Values of I(h, /) are calculated for polar angles between 0° and 11°. For the filaments with sinusoidal surfaces, the solution requires a summation over a series of Bessel functions. The general shape of the intensity distribution is obtained with two terms [i"2 in eq. (9)]. Higher order terms act only to minimize the lower amplitude oscillations that are caused by truncating the series. Similarly, the
Fig. 3. Iso intensity contour plots demonstrating the influence of parameters in Fig. 1 on the calculated two—dimensional SALS patterns. Varying amplitude of the disturbance at a fixed wavelength (a, b, c), varying wavelength at a fixed amplitude (c, d) and the resulting pattern for an array of spheres (e).
Droplet break-up in emulsions and polymer blends
smoothness of the aligned sphere model [eq. (16)] is determined by the number of spheres. Figure 3 shows iso-intensity plots of typical scattering patterns calculated with the diffraction models for filaments with sinusoidal surfaces (a, b, c, d) and aligned spherical droplets (e). The first three plots (a, b, c) demonstrate the effect of varying the amplitude of the disturbance at constant wavelength, while a comparison of plots (c) and (d) illustrates the change in scattering pattern with different wavelengths at a constant amplitude. For a"0 the scattering pattern is that of a cylinder (R "0.5 km), appearing as a streak 0 that is oriented perpendicular to the orientation direction of the filament. When a becomes non-zero, secondary streaks appear parallel to the primary streak. The intensity of these streaks depends on the value of a while their position is determined by the value of the wavelength j . As mentioned before, the amplitude R growth of the disturbance is bounded by a+0.82R 0 above which value the diffraction model for aligned spheres must be used. The smooth transition between the two models is demonstrated by the last two plots in Fig. 3. Plot (d) corresponds to a filament near breaking, with a"0.8R , while plot (e) shows the 0 predicted scattering for a series of aligned spheres. The influence of the different physical parameters is observed more easily by comparing the intensity measured along the /"0 axis, i.e. the intensity parallel to the orientation of the original filament. The amplitude of the disturbance is related directly to the amplitude of the intensity in the secondary streak [see Fig. 4(a)], while Fig. 4(b) demonstrates that varying the wavelength at a fixed value of a influences the position h of the secondary streak. As the wavelength is inm creased, the streak moves to smaller scattering angles through the relation kj sin h "2n [see eq. (10)]. R m The ability to clearly separate the influence of these two physical parameters on the features of the SALS patterns makes it possible to follow the development of the surface disturbance via SALS. If the forma-
2235
tion of the satellite droplets is neglected, the spacing between the aligned spherical drops is fixed by the wavelength of the disturbance through j " R 2R #s. The intensity of the secondary streak is $301 then controlled by the relative magnitude of R with respect to s. Since the formation of the $301 and satellite droplets is neglected, the values of R $301 s of the spheres can be deduced from R and the 0 wavelength of the disturbance. Therefore, the position of the secondary streaks after the filament breaks may be measured to investigate the final state of the morphology. The relationship kj sin h "2n provides a quick R m method for estimating the magnitude of the wavelength of the disturbance in real units. Unfortunately, the wave number includes the refractive index, which is often difficult to determine for a concentrated blend system. However, it is constant within a single blend, so the relationship is still quite powerful when comparing the results of different flow histories. Combining the two models discussed above, predictions can be made of the SALS patterns resulting from the development of Rayleigh instabilities on long filaments. The Tomotika analysis can be used to study break-up after cessation of shear flow, at least in the case of purely Newtonian fluids. The amplitude is expected to grow at constant wavelength and consequently the secondary streaks should display an increasing light intensity but should not change their position. 3. EXPERIMENTAL
3.1. Materials The materials used in the present investigation are polydimethylsiloxane (PDMS) from RhoˆnePoulenc (Rhodorsil 47V200.000) and polyisobutene (PIB) from Exxon Chemical (Parapol 1300). The steady state rheological properties of both materials have been measured with a Rheometrics Mechanical
Fig. 4. Calculated SALS intensity along the /"0 axis for the parameters used in Fig. 3. Part (a) is for fixed j "6 km with amplitude varying from a smooth cylinder to near breaking. Part (b) is for fixed amplitude R and changing j as well as a series of aligned spheres R "0.91 km and s"2.18 km. R $301
J. Mewis et al.
2236
Table 1. Material properties of the blend components
Polymer
Density (kg/m3)
n
g o (Pa s)
( 1,0 (Pa s2)
PDMS PIB-1300
975 895
1.406 1.499
200 90
10 0.06
Note: All values reported at 23°C.
Spectrometer RMS800 using cone and plate geometry (radius 12.5 mm, cone angle 0.1 rads) at 23.0°C (Table 1). The maximum shear rate in these experiments is limited by shear fracture, especially for the more elastic PDMS sample. The viscosities of the two materials are constant over the accessible range of shear rates. The PDMS is about twice as viscous as the PIB. Both polymers display some degree of elasticity and behave approximately as second-order fluids, characterized here by the first normal stress coefficient (( ). This 1,0 factor is about two decades higher for PDMS than for PIB. Blends consisting of mixtures of 1% by weight of PDMS in PIB are prepared using a turbine mixer with a six flat-blade disk. The freshly mixed blend is placed in a vacuum oven at room temperature to eliminate air bubbles. The interfacial tension ! between the two phases in the PDMS/PIB-1300 system is 3.1 mN/m (Sigillo et al., 1997) and the viscosity ratio p+2 at 23°C. 3.2. Experimental procedure For the system under investigation the droplet size is governed by the shear history, hence flow can be used to generate a well—defined initial condition for transient experiments. In the present experiments, a preshear at a shear rate c5"2 1/s is applied until the scattering pattern does not change with further shearing, in this case, after approximately 4000 strain units. The shear rate is then increased suddenly to 30 1/s for a short time. The preshear at 2 1/s acts to break up large droplets, but is not strong enough to greatly deform the smaller droplets. Based on previous work (Vinckier et al., 1996), it is known that the steady-state morphology is made up of slightly deformed droplets of radius R corresponding to those defined by the critical value of capillary number, Ca,g c5R/!, with m g the matrix viscosity and c5 the shear rate. The size of m the droplets is extracted from SALS measurements providing a radius of 2.5$0.5 km (Yang et al., 1997). After increasing the shear rate suddenly, the droplets deform affinely (Elemans, 1989) to high levels of l/R o so that, when the flow is stopped the morphology consists of extended fibrils of PDMS in a quiescent matrix of PIB. 3.3. Small-angle light scattering The flow small-angle light scattering apparatus (Fig. 5) used in this work is a modification of a Rheometrics Optical Analyser (ROA). It has a polarized 10 mW He—Ne laser light source (j"633 nm).
Fig. 5. Modified Rheometrics ROA used to capture SALS patterns of the blend system.
In these experiments, the polarization direction is continuously rotated by a half-wave plate (rotating at 2000 rps) before it reaches the sample cell. The sample is sheared in a parallel plate shear cell consisting of two optical grade glass plates. The upper platen of the cell is driven by a stepper motor with an angular precision of 25,000 steps per revolution. The beam passes through the sample along the velocity gradient direction at a distance of 0.53 cm from the centre of rotation. A sample thickness of 200 km is maintained to limit multiple scattering in these fairly turbid samples. The light scattered by the samples is visualized on a semi-transparent plane parallel to the velocity—vorticity plane, positioned at 4.95 cm below the cell. The maximum accessible scattering angle is about 15°. A circular beam stop is used to block the main beam and to avoid saturation of the camera. A color CCD camera (Ikegami model ICD-810P) is mounted in the optical track to record the transient scattering patterns. The patterns are either captured on video tape or directly stored on an IBM-PC, which is equipped with a frame grabber (Data Translations DT3851), for further analysis of the scattering patterns.
4. RESULTS AND DISCUSSION
Figure 6 shows the evolution of experimental SALS patterns for 1% PDMS in PIB during the shear and relaxation experiments. The sample is presheared at 2 1/s and then the shear rate is increased suddenly to 30 1/s for 4 s to create highly elongated filaments that are oriented close to the shear direction. The flow is
Droplet break-up in emulsions and polymer blends
2237
Fig. 6. Measured SALS patterns for 1% PDMS/PIB after a two-step shearing (preshear at c5"2 1/s, step up to 30 1/s for 4 s, then relaxation).
then stopped and the interface created by the extension of the inclusions is allowed to relax by way of surface instabilities. In the SALS patterns, a diffuse ellipsoidal scattering pattern is observed during the preshear at 2 1/s, with the major axis perpendicular to the flow direction. It reflects the presence of slightly deformed droplets. Upon increasing the shear rate suddenly to 30 1/s, the pattern distorts rapidly into a bright streak perpendicular to the flow direction, indicating that the high shear rate effectively stretches the droplets in the flow direction. Once the flow is stopped, streaks grow parallel to the primary streak in the SALS pattern. These secondary streaks increase in intensity during several seconds and then remain unchanged even for hours. The development of the SALS patterns after flow demonstrates that droplet retraction does not dominate the relaxation mechanism. Retraction would cause the scattering pattern to become circular rather than showing multiple streaks after long periods at rest. In Fig. 7 the evolution of the intensity along the /"0 axis is taken from the two-dimensional patterns in Fig. 6. From the previous section it is known that the position h of the streak is related to the m wavelength of the disturbance which causes the filaments to break up. At long times the position of h reflects the spacing of the resulting aligned spherim cal droplets. The variation of intensity in the streak is indicative of the growth of the amplitude of the disturbance and the formation of aligned spheres. Because the secondary streaks are initially weak in intensity, it
is impossible to determine if the position of the streaks varies immediately after stopping the flow. In this experiment, the appearance of the streaks, observed as peaks in Fig. 7, can be seen to clearly appear at 0.5 s after the flow is stopped. The diffraction model for a filament with an oscillating surface allows for an independent measure of j and the growth rate of the disturbance a(t) . From R the model, a relation between the position h of the m streak and the wavelength is obtained: h "arm csin(2n/kj ). In the experimental intensity profile R (Fig. 7), sharp peaks are not evident. This is due to a number of factors including background scatter and the fact that there will be a distribution of filament diameters resulting from a polydisperse distribution of initial droplet sizes. Although it is possible to account for background scatter and develop a model to account for polydispersity it should be noted that the weak peak in Fig. 7 may be picked out with far more accuracy than trying to extract size information from a monotonic intensity distribution, such as the diffraction pattern resulting from spheres, ellipsoids or cylinders. Therefore, in this work we consider only the peak position and relate it to an average filament diameter. For this system, the secondary peak in I(h, /"0) is observed at 6.0$0.4o, hence the value of j of the disturbance for these conditions corresponds R to 4.0$0.3 km. For a given dilute dispersion, consisting of a Newtonian fluid in another Newtonian fluid, Tomotika (1935) calculated the wavelength of the dominant disturbance which has the fastest growth
2238
J. Mewis et al.
Fig. 7. Intensity profiles along /"0 (from the SALS patterns in Fig. 6) during break-up of the fibrils. The maximum at h corresponds to the position of the secondary streaks. m
rate and eventually causes the filaments to break. The relationship between the dominant wavelength and initial filament radius is determined by the viscosity ratio of the dispersion (Tomotika, 1935; Mikami et al., 1975; Janssen, 1993). In the case of p"2, one finds R "0.52j /(2n) . This may be combined with the 0 R,m measured value of j "4.0 km to provide an estiR,m mate of the initial radius of the filament; i.e., R "0.3 km. A simple mass balance provides the size o and spacing of the resulting string of independent droplets after break-up. For the given flow history, the final morphology is a series of drops of R "0.7 km evenly spaced at s"2.6 km apart. $301 This is a direct measure of the effectiveness of droplet dispersion by shear and Rayleigh instabilities. A single large droplet created at a rate of 2 1/s is broken up into a number on the order of 100 smaller droplets. Inclusions of an initial radius of several micrometers are dispersed into droplets of radius 0.7 km by shearing at a high rate for only 4 s. Assuming that the initial inclusions are deformed affinely, the filament radius at cessation of flow may be estimated and compared to the one measured by SALS. Affine deformation implies that there is a direct relationship between the dimensions of the filament and the applied strain. The conditions of the flow history used here, shearing for 4 s at 30 1/s result in 120 strain units. Combined with the estimate for R derived from the SALS pattern, 2.5 km, the filament is calculated to be 612 km long. A simple volume balance of the inclusion then gives a filament radius of R "0.2 km. This value agrees reasonably well with 0 the value of 0.3 km determined from the dominant wavelength of the instability. The value determined through affine deformation strongly depends on the size of the initial inclusions which is known only approximately. The value of R determined by ap0 plying the model to the SALS pattern comes from a direct morphological measurement and depends only
on the viscosity ratio of the two materials p, and the assumptions of the model developed by Tomotika. Therefore, the value of 0.3 km is considered more accurate. The growth rate of the instability may be determined from the temporal increase in intensity of the streak. This should follow an exponential dependence as predicted by Tomotika (1935). However, the high level of background scattering in this system, due to turbidity, makes this comparison pointless at this time. In a more ideal system extracting structural information from the intensity growth in the streaks is feasible. One should also note that eq. (13) may be solved for any shape to describe the scattering of an array of objects. For example, an analysis of the case of aligned ellipsoids should provide the scattering from the system at the breaking point. This will improve the continuity in the calculation of evolution of the intensity of the secondary streaks patterns but will not change the position of the peaks. As we analyse only the peak position in this study, this refinement to the scattering models has not been included. It should be clear that the proposed technique is more accurate than attempting to determine the radius of the filaments directly from SALS patterns at high shear rates. The size of the filaments (0.1—1.0 km) falls between scattering theories (RGD, Mie and diffraction). The scattering function from a filament (cylinder) is a decaying function of scattering angle and is difficult to fit without considerable error and ambiguity. The appearance of a streak is therefore a much more accurate way of extracting size information from small-angle light scattering. The combination of well designed flow experiments with a series of scattering models provides a powerful technique for probing the morphology. Turbidity limits the application of the technique to low volume fractions of the dispersed phase. Yet, it can be used to study the effect of parameters such as viscoelasticity or the presence of compatibilizer in dilute systems of polymers used in real blends. Experiments on model systems more conducive to scattering are possible (Van Puyvelde et al., 1998) and are being used to elucidate the role of various system parameters on filament break-up.
5. CONCLUSIONS
Combining SALS data from a well-defined shear experiment with suitable diffraction models is shown to result in a viable technique for characterizing specific morphological features of a two-phase polymer blend during flow and in particular during filament break-up. Scattering models based on Fraunhofer diffraction provide predictions of the scattering patterns resulting from extended filaments, filaments with a sinusoidal surface disturbance and a series of aligned spheres. The combination of the latter two models has led to a prediction of the SALS patterns observed during break-up due to Rayleigh instabilities. Analysis of the scattering models makes it
Droplet break-up in emulsions and polymer blends
possible to quantify separately the effects of wavelength development and disturbance growth. They can be related to measurable scattering properties. An important result is the simple relationship (h "arcsin(2n/kj )) between the position of the m R streak in the SALS pattern and the magnitude of the dominant wavelength of the surface disturbance. The materials and scattering system utilized in this work limited the investigation to low concentrations and simple verification of the technique. Use of a more accurate SALS device will allow the wavelength and growth rate of the instabilities to be probed in situ and, hence, compared to mechanical measurements of flow properties. The influence of compatabilizers, viscoelasticity and interfacial tension on the break-up phenomenon are then easily investigated. Although this technique is limited to dilute systems, the ability to probe the influence of these parameters in situ will provide vital understanding of the filament break-up process. This technique is not restricted to polymer blends, but may be applied to emulsions and other two phase systems. The present work results in a more accurate technique for determining filament radius and offers the potential for characterizing filament break-up in model blend systems. The analysis of specific features of the SALS pattern, i.e. a streak, provides a much more accurate technique for extracting length scale information from these patterns than using the monotonically decaying part of the pattern. A carefully designed experimental protocol allows morphological size information to be extracted during flow and break-up. The effect of material parameters (p, !, g ) m on the flow-induced morphology can be probed and characterized rapidly in situ with this procedure.
Acknowledgements Partial financial support for this project has been provided by the European BRITE-EURAM program (Contract No. BRE2.CT92.0213), Onderzoeksfonds K.U. Leuven and FKFO. LMW acknowledges the National Science Foundation (INT-9505545) for post-doctoral funding of this work.
REFERENCES
Bentley, B. J. and Leal, L. G. (1986) An experimental investigation of drop deformation and breakup in steady, two-dimensional linear shear flows. J. Fluid Mech. 167, 241—283. Elemans, P. H. M. (1989) Modelling of the processing of incompatible polymer blends. Ph.D. thesis, Eindhoven University of Technology. Elemans, P. H. M., Bos, H. L., Janssen, J. M. H. and Meijer, H. E. H. (1993) Transient Phenomena in Dispersive Mixing. Chem. Engng Sci. 48, 267—276. Fan, Y. Y. and Huynh, V. M. (1993) Light scattering from periodic rough cylindrical surfaces. Appl. Opt. 32, 3452—3458. Grace, H. P. (1982) Dispersion phenomena in high viscosity immiscible fluid systems and application of static mixers as dispersion devices in such systems. Chem. Engng Commun. 14, 225—277.
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Janssen, J. M. H. (1993) Dynamics of liquid/liquid mixing. Ph.D. thesis, Eindhoven University of Technology. Janssen, J. M. H. and Meijer, H. E. H. (1993) Droplet breakup mechanisms: stepwise equilibrium vs transient dispersion. J. Rheol. 37, 597—608. Korn, G. A. and Korn, T. M. (1961) Mathematical Handbook for Scientists and Engineers. McGrawHill, New York. Mikami, T., Cox, R. G. and Mason, S. G. (1975) Breakup of extending liquid threads. Int. J. Multiphase Flow 2, 113—138. Rallison, J. M. (1984) The deformation of small viscous drops and bubbles in shear flows. Ann. Rev. Fluid Mech. 16, 45—66. Sethuraman, J. (1986) Optical display of the ambiguity function and the diffraction pattern of a screw—an analogy. Am. J. Phys. 54, 450—452. Sigillo, I., di Santo, L., Guido, S. and Grizzuti, N. (1997) Comparative study of interfacial tension in a model polymer blend. Polym. Engng Sci. 37, 1540—1549. Sondergaard, K. and Lyngaae-Jorgensen, J. (1996) Coalescence in an interface modified polymer blend revealed by light scattering measurements. Polymer 37, 509—517. Sondergaard, K., Valenza, A. and Lyngaae-Jorgensen, J. (1992) Effect of a compatibilizer on the morphology of polymer blends revealed by light scattering measurements during flow. Polym. Network Blends 2, 139—165. Stone, H. A. (1994) Dynamics of drop deformation and breakup in viscous flows. Ann. Rev. Fluid Mech. 26, 65—102. Stone, H. A. and Leal, L. G. (1989) Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid. J. Fluid Mech. 198, 399—427. Stone, H. A., Bentley, B. J. and Leal, L. G. (1986) An experimental study of transient effects in the breakup of viscous drops. J. Fluid Mech. 173, 131—158. Taylor, G. I. (1932) The viscosity of a fluid containing small drops of another fluid. Proceedings of Royal Society of ¸ondon A 138, 41—48. Taylor, G. I. (1934) The formation of emulsions in definable fields of flow. Proceedings of Royal Society of ¸ondon A 146, 501—523. Tomotika, S. (1935) On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proceedings of Royal Society of ¸ondon A 150, 322—337. Torza, S., Cox, R. G. and Mason, S. G. (1972) Particle motions in sheared suspensions: XXVII. Transient and steady deformation and burst of liquid drops. J. Colloid. Interface Sci. 38, 395—411. van de Hulst, H. C. (1957) ¸ight Scattering by Small Particles. Dover, New York. Van Puyvelde, P., Yang, H., Mewis, J. and Moldenaers, P. (1998) Rheo-optical probing of relaxation phenomena in immiscible polymer blends. J. Colloid Int. Sci. 200, 86—94. Vinckier, I., Moldenaers, P. and Mewis, J. (1996) Relationship between rheology and morphology of model blends in steady shear flow. J. Rheol. 40, 613—631. Yang, H., Moldenaers, P. and Mewis, J. (1998) Rheooptical investigation of immiscible polymer blends, Polymer (in press).