3740
Langmuir 2000, 16, 3740-3747
On the Existence of a Stress-Optical Relation in Immiscible Polymer Blends P. Van Puyvelde,* P. Moldenaers, and J. Mewis Department of Chemical Engineering, K. U. Leuven, de Croylaan 46 B-3001, Heverlee, Belgium
G. G. Fuller Department of Chemical Engineering, Stanford University, California 94305 Received October 26, 1999. In Final Form: January 14, 2000 In emulsions and immiscible polymer blends linear conservative dichroism is known to be very sensitive to flow-induced microstructural changes during flow. The same holds for the excess or interfacial contribution to the first normal stress. Similarities have already been observed between these two properties in blends. Here it is investigated whether this similarity reflects a more quantitative relationship. The interfacial normal stress can be described by means of an interface anisotropy tensor. Therefore, the measured dichroism will be compared with the calculated components of this anisotropy tensor for different flow histories. These include steady-state shear flow, sudden increase in shear rate, relaxation with droplet breakup, and oscillatory flow. In all cases a proportionality factor is obtained. For strongly deformed droplets direct theoretical evidence for a quantitative relationship between rheology and rheo-optics in blends is provided. To extend this relationship to a larger window of shear rates, the size dependence of the scattering has to be taken into account.
1. Introduction When two immiscible polymers or other fluids are mixed, a complex phase morphology develops.1 Due to an applied flow field the phases might deform, rupture, or recombine. These structural changes are reflected in the rheological behavior. Rheology has proven to be very suitable to follow in situ the development of the morphology during flow.2-7 In particular, the first normal stress difference (N1) turns out to be a very sensitive probe of the structure. More recently, linear conservative dichroism (∆n′′) has been shown to be another powerful tool to investigate the flow-induced morphological changes in emulsions and twophase polymer blends.8-11 The evolution of the dichroism and of the first normal stress difference show similar features: e.g., after a step-up in shear rate, both N1 and ∆n′′ initially increase, reach a maximum, decrease, and finally evolve toward a steady-state value.5,8 The same structural interpretation, based on deformation, breakup, and coalescence of the dispersed phase can be used to explain their time evolution. In relaxation experiments either N1 or ∆n′′ has been used to identify the different relaxation mechanisms such as droplet retraction, endpinching, and Rayleigh instabilities.6,11 In addition, similar (1) Doi, M. In Lecture Notes in Physics; Garrido, L., Ed.; SpringerVerlag: Berlin, 1992. (2) Takahashi, Y.; Kitade, S.; Kurashima, N.; Noda, I. Polymer J. 1994, 26, 1206. (3) Takahashi, Y.; Kurashima, N.; Noda, I.; Doi, M. J. Rheol. 1994, 38, 699. (4) Vinckier, I.; Moldenaers, P.; Mewis, J. J. Rheol. 1996, 40, 613. (5) Vinckier, I.; Moldenaers, P.; Mewis, J. J. Rheol. 1997, 41, 705. (6) Vinckier, I.; Moldenaers, P.; Mewis, J. Rheol. Acta 1997, 36, 513. (7) Guenther, G. K.; Baird, D. G. J. Rheol. 1996, 40, 1. (8) Yang, H.; Zhang, H.; Moldenaers, P.; Mewis, J. Polymer 1998, 39, 5731. (9) Van Puyvelde, P.; Yang, H.; Mewis, J.; Moldenaers, P. J. Colloid Interface Sci. 1998, 200, 86. (10) Vermant, J.; Van Puyvelde, P.; Moldenaers, P.; Mewis, J.; Fuller, G. G. Langmuir 1998, 14, 1612. (11) Van Puyvelde, P.; Moldenaers, P.; Mewis, J. Phys. Chem. Chem. Phys. 1999, 1, 2505.
scaling relations have been found to hold for both quantities. The above-mentioned observations lead to the suggestion that a relation could exist between the stress tensor and the imaginary part of the refractive index tensor, i.e. the dichroism. A stress-optical law has been shown to hold between the real part of the refractive index tensor or birefringence and the stress tensor for polymer melts and polymer solutions, as discussed in the review of Janeschitz-Kriegl.12 The physical basis for this law is that both the birefringence and the entropic contribution to the stress tensor are related to the second-order moment of the distribution for the end-to-end vector. A stressoptical relation involving conservative dichroism rather than birefringence has been demonstrated to hold for suspensions of Brownian hard spheres.13 Bender and Wagner used it to separate the effects of thermodynamic and hydrodynamic contributions to the stress tensor. Separating different types of stresses has also been performed by van Egmond,14 who derived a stress-optical relation between dichroism and stresses due to concentration fluctuations in polymer solutions in rather poor solvents. In two-phase liquid/liquid mixtures the interface causes an excess stress. It can be related to the orientation distribution of the interface. Doi and Ohta applied this to model the rheology of immiscible blends.15 In systems with a droplet/matrix structure the linear conservative dichroism is also affected by the shape and the orientation of the inclusions. In this work, we will investigate how close these excess stresses and dichroism are related in the case of two-phase liquids. (12) Janeschitz-Kriegl, H. In Polymer Melt Rheology and Flow Birefringence; Springer-Verlag: Berlin, 1983. (13) Bender, J. W.; Wagner, N. J. J. Colloid Interface Sci. 1995, 172, 171. (14) van Egmond, J. Macromolecules 1997, 30, 8045. (15) Doi, M.; Ohta, T. J. Chem. Phys. 1991, 95, 1242.
10.1021/la991406w CCC: $19.00 © 2000 American Chemical Society Published on Web 03/18/2000
Stress-Optical Relations in Immiscible Polymer Blends
2. Materials and Methods A model polymer blend has been used consisting of 3% polybutadiene (PB, Mw ) 4500 from Janssen Chimica) dispersed in polyisobutene (PIB, Parapol 950 from Exxon). The components are immiscible and liquid at room temperature, thus avoiding experimental problems, such as thermal degradation, that are often encountered in molten polymer blends. This model blend has a viscosity ratio (p ) droplet viscosity over matrix viscosity) of 0.27 at 23 °C and the interfacial tension, as obtained by a rheo-optical breaking-thread method, is approximately 0.2 mN/ m.9 The linear conservative dichroism experiments have been performed on a Rheo-Optical Analyzer (ROA, Rheometric Scientific). The instrument uses a modulation technique of the polarization state of the light (wavelength of the light λl is 632.8 nm).16 Both parallel plates and a Couette cell can be used as flow geometry. Details of the experimental setup can be found elsewhere.8 To measure dichroism under oscillatory flow the ROA has been equipped with a servomotor. The linear dynamic moduli, G′(ω) and G′′(ω), have been obtained on a Dynamic Stress Rheometer (Rheometric Scientific).
Doi and Ohta15 proposed a constitutive equation for a 50/50 mixture of two immiscible Newtonian liquids having identical viscosities. In their phenomenological approach, the presence of an interface is represented by two variables. The first expresses the total amount of interface per unit volume (Q), the second describes the shape and orientation of this interface by means of a second-order tensor which is called the interface tensor. The components of q are given by:17
1 V
∫ dS (ninj - 31 δij)
(1)
with V the volume of the sample, S the total interfacial area, and ni the components of a unit vector locally normal to the interface. The details of the morphology of the dispersed phase (i.e., particle size, shape, orientation, distributions, etc.) are not explicitly taken into account. Roughly speaking, the interface tensor is determined by two factors: one is the flow field which tends to enlarge and orient the interface and the other is the interfacial tension between the components of the blend which opposes these effects. The resulting total stress tensor is given by:
σij ) η(dij + dji) - Γqij - pδij
(2)
with dij the components of the velocity gradient tensor and Γ the interfacial tension. The first term represents the bulk contribution of the Newtonian fluids to the stress assuming a linear, volume based mixing rule. The second term in eq 2 contains the contribution to the stress due to the interfacial tension. Due to the presence of an interface, excess stresses can be generated yielding the expressions (1 ) flow direction, 2 ) gradient direction):
N1,excess ) -Γ(q11 - q22) σ12,excess ) -Γq12
dynamically consistent transport equations for the multiphase flow of equidensity, equiviscous Newtonian fluids with a constant interfacial tension. Equation 3 indicates that the interface tensor can be used to describe the excess stresses if the components of q can be calculated for the actual morphology. 3.1. Droplet Deformation During Flow. Vinckier et al.15 calculated the components of q for a disperse phase undergoing a sudden, large increase in shear rate. They assumed the droplets to be cylinders with hemispherical ends. In that case the components qij (eq 1) can be related to the angle θ between the cylinder axis and the flow direction:
q12 ) -
(3)
The interfacial terms link the rheology to the microstructure of these complex fluids. Wagner et al.18 demonstrated that the Doi-Ohta model is a set of thermo(16) Fuller, G. G. In Optical Rheometry of Complex Fluids; Oxford University Press: Oxford, U.K., 1995. (17) Batchelor, G. K. J. Fluid Mech. 1970, 41. (18) Wagner, N.; Ottinger, H. C.; Edwards, B. J. AIChE J. 1999, 45, 1169.
φ cos θ sin θ r
(4)
φ 2 sin2 θ r 3
( φ ) (cos r
q11 ) q22
3. Time-Dependent Behavior
qij )
Langmuir, Vol. 16, No. 8, 2000 3741
2
) 2 θ- ) 3
(5) (6)
with φ the volume fraction of the dispersed phase and r the radius of the cylinder. According to eq 3, excess stresses are then given by:
Γφ cos θ sin θ r
(7)
Γφ (cos2 θ - sin2 θ) r
(8)
σ12,excess ) N1,excess )
For a sufficiently large step-up ratio in shear rate, the deformation can be assumed to be approximately affine.9 Under these conditions, the dimensions of the deforming cylinder can be expressed as a unique function of strain (γ). Equations based on the combination of affine deformation and the interface tensor have been found to be capable of describing the excess first normal stress difference during step-up experiments in model blends:5
(
Γφ 1 + σ12,excess )
(
γΓφ 1 + N1,excess )
)
γ2 γ + x4 + γ2 2 2
1/4
Rxγ2 + 4
(9)
)
γ2 γ + x4 + γ2 2 2
1/4
Rxγ2 + 4
(10)
with R the radius of the initial droplet. For large strains the angle θ becomes very small and the excess stresses can therefore be evaluated as:
σ12,excess ) 0 N1,excess )
(
(11)
)
Γφ γ2 γ 1 + + x4 + γ2 R 2 2
1/4
(12)
More accurate descriptions of the shape of a deforming droplet are available but are not essential in the present discussion.20 The evolution of the dichroism during similar flow conditions has been modeled by Yang et al.8 assuming affine deformation and Rayleigh-Gans-Debye scattering. For a dilute dispersion of dielectrically isotropic nonab(19) Elemans, P. H. M. Ph.D. Thesis, T. U. Eindhoven, 1989. (20) Maffettone, P. L.; Minale, M. J. Non-Newtonian Fluid Mech. 1998, 78, 227.
3742
Langmuir, Vol. 16, No. 8, 2000
Van Puyvelde et al.
ponents of the stress tensor can be written as:
σ13 ) 0 σ11 - σ33 )
(16) ∆n′′ R
(17)
As the cylinders are assumed to be axisymmetric and their orientation angle θ is negligibly small, the two interfacial excess normal stress differences are equal:23
N1,excess ) σ11 - σ22 ≈ σ11 - σ33
Figure 1. Difference in scattering efficiency as function of the radius of the stretched filament.
Taking into account eq 18, the combination of eqs 12 and 14 leads to the following relation between dichroism and the excess first normal stress difference:
∆n′′ )
sorbing particles, the dichroism can be written as:21
∆n′′ )
∑ (Csca,1 - Csca,2) 2k i)1
λlφ 2
2nmπ R
λl 2nmπ2Γ
N1,excess
(19)
N
1
(13)
where N is the number of particles per unit volume, k ) 2πnm/λl the wavenumber of the light (nm being the refractive index of the matrix) and Csca,i the scattering cross sections according to two orthogonal polarization directions in the plane perpendicular to the direction of the incident light. The scattering cross sections depend on size and shape of the inclusions. To avoid different scattering approximations for different sizes, a numerical procedure is followed based on the direct solution of Maxwell’s equations.22 Assuming fiberlike inclusions, affine deformation and a strain that is large enough to have an orientation angle close to zero, the following expression is obtained for the evolving dichroism:8
∆n′′ )
(18)
(
1+
)
γ2 γ + x4 + γ2 2 2
1/4
(14)
Here, is the difference in scattering efficiency (Csca,1 Csca,2)/2rl, l being the length of the fibrils. It is plotted as a function of fiber radius r for the system under consideration in Figure 1. It can be seen to be approximately constant for micron-sized inclusions. If a relation exists between the imaginary part n′′ of the refractive index tensor and the interfacial contribution to the stress tensor σ is assumed, the components of the stress tensor should be related to the difference in eigenvalues of n′′. For a shear flow in the (1,2)-plane and light propagating along the 3-axis (e.g., a Couette geometry), this would lead to the following equation:
σ12 )
∆n′′ sin 2χ′′ 2R
σ11 - σ22 )
∆n′′ cos 2χ′′ R
(15)
with R a stress optical coefficient, χ′′ the angle of dichroism defined as the angle between the major axis of nij and the flow direction. If the dichroism measurements are performed in the 1-3 plane (parallel plates), the orientation angle of the dichroism is zero and the interfacial com(21) Van der Hulst, H. C. In Light Scattering by Small Particles; Dover Publications: New York, 1958. (22) Barber, P. W.; Hill, S. C. In Light Scattering by Particles: Computational Methods; World Scientific: London, 1990.
Equation 19 suggests the value of the stress-optical coefficient R to be given by:
R)
λl 2nmπ2Γ
(20)
It contains the interfacial tension and refractive index of the matrix phase as material parameters and depends on droplet size through the difference in scattering efficiency , which also depends on the relative refractive index ratio m. Combining eqs 3 and 19 links the dichroism with the components of q:
∆n′′ ) -
λl 2nmπ2
(q11 - q22) ) - C(q11 - q22) (21)
This relation between dichroism and the interface tensor is an important result because it provides a viable alternative of modeling the dichroism under various flow conditions. Although a clear size dependence is present in the proposed expression for the stress-optical coefficient through the parameter , it can be approximated by a constant value in the region of cylinder radii relevant to this problem. This is illustrated in Figure 1 where has a value of 0.00 056 ( 0.00 012. It leads to a calculated value for C of 1.2 × 10-11 ( 0.2 × 10-11 for the present system. The linear relation between ∆n′′ and (q11 - q22) suggested by eq 21 has been used to describe the dichroism response for step-up experiments with the coefficient C as a fitting factor. The result is shown in Figure 2 where the transient dichroism for three step-up experiments are compared with eq 21. It can be seen that eq 21 describes rather accurately the deformation stage of the droplets during these experiments whenever the step-up is large enough to ensure affine deformation. The maximum in dichroism reflects the breaking up of the filaments, which reduces their anisotropy. The fitted value for C is 1.15 × 10-11, in agreement with the theoretically derived value. Deviations occur at low shear rates because the affinity assumption fails below a critical value of shear rate or capillary number.19 3.2. Fibril Breakup by Rayleigh Instabilities. It is known that fibrillar structures can break up under quiescent conditions due to Rayleigh instabilities. Vinckier (23) Onuki, A. Int. J. Thermophys. 1995, 16, 381.
Stress-Optical Relations in Immiscible Polymer Blends
Langmuir, Vol. 16, No. 8, 2000 3743
with q11(t) - q22(t) now given by eq 23. Equation 24 implies that, when the flow is stopped, the orientation angle of the droplets remains negligibly small and that the droplets are axisymmetric, hence that eq 18 is still valid. As an example, in Figure 3 experimental results are fitted to eq 24. The theoretical description ends when the filament breaks. The tail of the relaxation curve is caused by the retraction of the small droplets that result from the breakup of the fibril. This process is not considered here. It can be seen that the experiment can be fitted quite well by eqs 23 and 24. The fitting produces a value of 1.12 × 10-11 ( 0.05 × 10-11 for C. It is in good agreement with that of 1.15 × 10-11 deduced from step-up flows and the theoretical value of 1.2 × 10-11. Figure 2. Evolution of the dichroism during step-up experiments (step-up ratio O: 5, 0: 6, 4: 10). The solid line corresponds to the theoretical prediction based on the anisotropy of the interface.
et al.6 pointed out that, when stopping the flow under conditions where fibrils are expected to break in this manner, the N1 relaxation curve shows a typical shoulder. These conditions can be induced by suddenly increasing the shear rate and then stopping the flow before the stretched filaments start to break during flow. Van Puyvelde et al.11 demonstrated that the dichroism relaxation describes similar curves when the same flow history is applied. Again a model for the behavior of a single droplet can be used to calculate the evolution of the q tensor and consequently the stress components according to eqs 2 and 3. The problem of a cylindrical Newtonian thread with a sinusoidal distortion superimposed on its surface, embedded in a quiescent Newtonian matrix, has been analyzed by Tomotika.24 Using his equations for the growth of the disturbances, the time evolution of the components of the interface tensor under the action of Rayleigh instabilities can be calculated:6,11
q11 - q22 )
2φ 3φ R0 R0
1
x (
)
2πR(t) 1+ λm
(22)
2
with R0 the radius of the fibril at the moment the flow is stopped, λm the dominant wavelength that will cause the breakup of the fibril (only a function of the viscosity ratio of the blend) and R(t) the time-dependent amplitude of the distortion. Experimentally, it has been found that the initial decay of both N1 and ∆n′′ is not described by eq 22. This was attributed to the occurrence of end-pinching which takes place before Rayleigh instabilities start to develop. It has been taken into account by an empirical correction factor: 6,11
q11 - q22 )
[
2φ 3φ R0 R0
1
x (
)
2πR(t) 1+ λm
]
exp
2
(-tτ )
(23)
with τ the experimentally determined relaxation time representing the intial decay. Equation 23 has been used by Vinckier et al. to model the relaxation of N1,excess.6 If there were a stress-optical relation, the dichroism would at each instant again be described by:
∆n′′(t) ) -C(q11(t) - q22(t)) (24) Tomotika, S. Proc. R. Soc. London A 1935, 322.
(24)
4. Steady-State Shear Flow In this section, an attempt is made to model dichroism in steady-state shear flow starting from the interfacial anisotropy tensor q. Dichroism measurements have been performed both in a parallel plate geometry and in a Couette geometry. In the former, the dichroism has to be compared with q11 - q33, whereas in the Couette geometry q11 - q22 is the relevant factor. To predict the components of the anisotropy tensor under steady-state conditions, a theory is needed that describes the deformation and the orientation angle as a function of the capillary number. Contrary to the procedure in Section 3, droplets under steady shear conditions cannot be modeled as long slender filaments. When an emulsion of noninteracting droplets of radius R with a total volume fraction of droplets φ is sheared, the inclusions deform into ellipsoids with a given shape and orientation. Both the components of q and the scattering caused by ellipsoids can be calculated but the relation between the two is not straightforward. Here, the calculated components of q will be compared with experimental dichroism data for steady-state shear flow. The deformed droplets are represented by a deformation parameter D ) (L - B)/(L + B), with L and B the long and short semi-axis of the droplet, and an orientation angle θ with respect to the flow direction. The components of the anisotropy of interface tensor for an ellipsoid are known.25 For the deformation parameter, the theoretical predictions proposed by Cox26 will be used here. Experimental data for the orientation angle are available at a few shear rates for our sample.10 For other shear rates it will be calculated with the Chaffey-Brenner theory which is known to yield a good description of the orientation angle in this system.10 The deformation according to Cox and the orientation angle according to Chaffey-Brenner are given by:
D)
L-B ) L+B
θ)
5(19p + 16) 4(p + 1)
x
20 (19p) + Ca 2
(25) 2
( )
π (19p + 16) (2p + 3) Ca 4 80(p + 1)
(26)
in which p is the viscosity ratio and Ca the capillary number, defined as the ratio of hydrodynamic stress (ηmγ˘ ) (25) Park, O. O.; Lee, H. M. Proc. 12th Ann. Meeting Polym. Proc. Soc. 1996, 191. (26) Cox, R. G. J. Fluid Mech. 1969, 37, 601.
3744
Langmuir, Vol. 16, No. 8, 2000
Figure 3. Comparison between the measured relaxation of the dichroism (open symbols) and the prediction of the model (solid line) after shearing for 3 s at 7 s-1.
Figure 4. Steady-state droplet size as determined from SALS patterns as a function of shear rate.
Figure 5. Steady-state dichroism (parallel plate geometry) and calculated anisotropy of the interface as a function of shear rate.
over the interfacial stress (Γ/R). The droplet size at a particular shear rate can be determined by means of light scattering experiments upon cessation of flow and an analysis based on the Debye-Bueche theory.8 The results are shown in Figure 4. An inverse proportionality between the droplet size and shear rate can be seen to apply over a limited range of shear rates. 4.1. Anisotropy and Stress-Optical Coefficient in the 1-3 Plane. In Figure 5, the dichroism measured in a parallel plate geometry is compared with the calculated anisotropy of the interface (q11 - q33) as a function of shear rate. The values of q11 - q33 are based on experimentally obtained values of the droplet radius up to a shear rate of 12 s-1 (see Figure 4). The values of q11 - q33 at higher
Van Puyvelde et al.
Figure 6. Proportionality factor between dichroism and the anisotropy of the interface as a function of shear rate. Open symbols: parallel plates. Closed symbols: Couette cell.
Figure 7. Scaled proportionality factor between dichroism and the anisotropy of the interface as a function of shear rate for different blend systems (O: PIB/PDMS (m ) 1.086), 0: PDMS/ PIB (m ) 0.926), 4: PB/PIB (m ) 1.01)).
shear rates are based on an extrapolation of the droplet radius assuming an inverse proportionality with shear rate. From the data of Figure 5 a proportionality factor between ∆n′′ and q11 - q33 can be calculated. Its value is shown in Figure 6 as a function of shear rate. It can be seen that the obtained proportionality constant C is quasi independent of shear rate up to a value of 12 s-1 yielding a value of approximately 1.2 × 10-11. The results for higher shear rates will be discussed below. Although under steady state flow conditions the morphology consists of ellipsoids oriented at a certain angle with respect to the flow direction, approximately the same value for C is obtained as in the morphology condition of long slender filaments (Section 3). Values of q11 - q33 have also been calculated for other blend systems. These include 1% PIB in PDMS (viscosity ratio ) 0.44) and 1% PDMS in PIB (viscosity ratio ) 2.3). The steady-state dichroism and droplet sizes for these blends have been reported by Yang et al.8 The resulting coefficients C are shown in Figure 7, together with the present data. The parameter that defines the optical difference between the three blend systems in Figure 7 is the relative refractive index m, i.e., the ratio of the refractive index of the droplet phase over that of the matrix phase. As dichroism is proportional to (m - 1)2,27 the coefficients C for the different systems can be scaled by using this factor. (27) Frattini, P. L.; Fuller, G. G. J. Colloid and Interface Sci. 1987, 119, 335.
Stress-Optical Relations in Immiscible Polymer Blends
Figure 8. Master curve of the normalized calculated dichroism.
For all three blends it can be observed that the curves of C versus shear rate have a similar shape: a rather constant value at low shear rates with a subsequent dropoff at higher shear rates. The shear rate at which C starts to decrease depends on the viscosity ratio p of the blend: the higher p, the higher this critical shear rate becomes. At low shear rates, when the product of the magnitude of the wave vector (k) with the minor semi-axis of the ellipsoid (B) is larger than approximately 2, the size dependence of the dichroism is relatively weak.27 This corresponds to the size range at which a rather constant value of C is observed. At higher shear rates, the droplet size decreases below the critical value and consequently size effects should be expected for the dichroism. The higher the viscosity ratio, the higher the shear rate that is required in order to deform the droplets sufficiently to reach the critical B, as is seen in Figure 7. To obtain a stress-optical relation that is valid over a wider range of shear rates an appropriate correction is required that takes into account the sensitivity of dichroism to size. For a given aspect ratio this depends on system parameters such as interfacial tension and viscosity ratio. In this discussion the correction factor will be calculated for a blend consisting of 3% PB in PIB. Considering the small droplet sizes involved, the Rayleigh-Gans-Debye approximation can be applied. Dichroism for an ellipsoid in this approximation can be readily calculated and is given by:28
∆n′′ )
φnmk3V(m - 1)2 8π2
×
∫02π ∫0π R(ϑ, φ)2 cos 2φ sin3 ϑdϑdφ]
[
(27)
where V is the volume of a particle. The function R(ϑ, φ) is given by:28
R(ϑ, φ) )
3 sin K(ϑ, φ) 3
K(ϑ, φ)
-
3 cos K(ϑ, φ) K(ϑ, φ)2
(28)
where K(ϑ, φ) is defined as 2k sin(ϑ/2) [(B2 + (L2 - B2) cos2 β)1/2] and cos β ) sin θ cos(ϑ/2) cos(ξ - φ) - cos θ sin(ϑ/2). The angles θ and ξ determine the orientation of the particle. The calculations are performed for the parallel plate geometry (1-3 plane) where ξ )90° and θ the orientation angle of the droplets with respect to the shear direction.28 The calculated dichroism is shown in Figure 8 for different sizes and aspect ratios of the 3% PB in PIB blend. (28) Meeten, G. H. J. Colloid Interface Sci. 1981, 84, 235.
Langmuir, Vol. 16, No. 8, 2000 3745
Figure 9. Stress-optical coefficient corrected based on calculations of the dichroism from ellipsoids (open symbols: parallel plates, closed symbols: Couette geometry). The dotted line indicates the lower limit for the application of the RGD theory.
For a fixed aspect ratio, the dichroism-size curve displays a maximum. In addition the maximum in dichroism always occurs at the same droplet radius for different aspect ratios, at least for aspect ratios relevant for steadystate flow. This maximum value can be used to normalize the calculated dichroism. As can be seen in Figure 8 this procedure results in a single master curve for the range of aspect ratios under consideration. The curve thus generated reflects the sensitivity of the dichroism to size. The range of shear rates where the coefficient C was found to be nearly constant (see Figure 6) corresponds to the size range around the maximum in Figure 8. In addition, the shape of the curve in Figure 8 is similar to the curve in Figure 6. Hence the normalized curve of Figure 8, or rather its inverse, can serve as a correction factor for C, at least for 1/R sufficiently large. Multiplying the experimental stress-optical coefficient with this correction function results in a coefficient which is constant over a wide range of shear rates (Figure 9). In principle, one should be able to use turbidity information to get the front factor in the Meeten calculation of the dichroism. However in practice this is not very accurate and the method presented here captures the size dependence of the stress-optical coefficient in more detail. 4.2. Anisotropy and Stress-Optical Relation in the Shear Plane. With a Couette or coaxial cylinder geometry the shear plane, i.e., the velocity-gradient plane can be investigated. A comparison between the anisotropy and dichroism in this plane is now more complicated because the orientation angle of the structure has to be considered. Combining eqs 3 and 15 one finds:
∆n′′ ) - C
(q11 - q22) cos 2χ′′
(29)
C has been determined here by using experimental values of ∆n′′ measured in the Couette cell and values for q11 q22 according to Park and Lee.25 The orientation angle χ′′ has been measured directly in the rheo-optical experiments.10 Although the number of data points is limited, the same trends as in the parallel plate geometry can be observed. At low shear rates C is approximately a constant, its value coinciding with that resulting from the parallel plate experiments. At higher shear rates C decreases. A size correction procedure, similar to the one derived for the parallel plate experiments, has also been applied to the Couette data. Dichroism calculations for different aspect ratios have again been performed based on the analysis by Meeten.28 After normalization a master curve
3746
Langmuir, Vol. 16, No. 8, 2000
Figure 10. Dynamic moduli after shearing at 1 s-1.
could be obtained, the inverse of which provides a correction for droplet size. This correction function for the Couette geometry is somewhat steeper in the size region of interest than that for the parallel plate geometry, indicating that the results in the former geometry are more sensitive to size. The corrected values of the proportionality constant in the Couette geometry are also displayed in Figure 9. Although the amount of data is rather limited, a rather constant value of C is obtained. 5. Stress and Dichroism Measurements under Oscillatory Shear Flow In the previous sections, flows have been considered in which droplets were deformed substantially. It remains to be verified whether the same stress-optical coefficient also describes the other extreme, i.e., a perturbation of the spherical shape. This can be achieved in small amplitude oscillatory flow. Even in the dilute system used here (3% PB in PIB) rheological measurements could be performed. This provides an opportunity for comparing the dichroism directly with the viscoelastic measurements rather than with the q tensor. It is known that the response to oscillatory flow depends on the droplet size. The latter can be controlled by submitting the samples for a sufficiently long time to steady shearing prior to the dynamic measurements. Figure 10 shows the linear dynamic moduli after shearing at 1 s-1. The components of the blend do not display any intrinsic elasticity and hence they do not contribute to the storage moduli G′. Therefore, the measured values of G′ reflect the elasticity induced by the interface. Because of the low value of the interfacial tension, the values of G′ are also quite low. G′ should reach a plateau at a frequency corresponding to the relaxation time of the droplets according to emulsion theory. Figure 11 shows the in-phase and out-of-phase components of the dichroism under the same experimental conditions. By applying different strains, it has been verified that the linear response is measured. At low frequencies, both components of the dichroism are of the same order of magnitude. At higher frequencies, the inphase contribution reaches a plateau whereas the outof-phase component still increases. The frequency at which the in-phase component reaches a plateau corresponds again to the relaxation time of the droplets. Experiments have been performed at different preshear rates, corresponding to different initial droplet sizes. Again, the size effect was observed both in the components of the modulus and the dichroism.
Van Puyvelde et al.
Figure 11. Dynamic moduli multiplied with stress-optical coefficient compared with in-phase and out-of-phase components of the dichroism (preshear rate 1 s-1).
To evaluate the stress-optical relation, the in-phase component of the dichroism should be compared with G′:
(∆n′′)′ )
C G′ Γ
(30)
Comparing the moduli in Figure 10 and the components of the dichroism in Figure 11 it can be observed that the measured values of (∆n′′)′ and G′ indeed evolve similarly. The two curves can be superimposed by using a suitable shift factor. The out-of-phase components should then be superimposed by the same shift factor. From the measured loss modulus however, the contribution of the components should be subtracted first to produce the excess contribution caused by the interface. In a dilute emulsion this means subtracting from G′′ the matrix contribution ηmω, where ηm is the matrix viscosity. The resulting curves are compared with the components of the dichroism in Figure 11. It can be seen that the two components of the modulus and those of the dichroism are superimposed with the same shift factor. The latter should be equal to C/Γ (eq 30). The value of C used in Figure 11 is 1.2 × 10-11, which is exactly the value derived theoretically as well as found experimentally in other experiments with different droplet geometries (see Sections 3 and 4). 6. Conclusions It has been demonstrated experimentally that the linear conservative dichroism is proportional to the anisotropy tensor which is known to be proportional to the excess stress tensor. The flow conditions used include steadystate shear flow, a sudden increase in shear rate, relaxation involving filament breakup by Rayleigh instabilities, and oscillatory flow. These conditions cover a wide range of shapes of the dispersed phase: from nearly spherical droplets over deformed ellipsoids to long filaments with an undulating surface. In each case, the same proportionality factor was obtained between shape anisotropy and dichroism. For cylindrical droplets oriented in the flow direction the proportionality between stresses and linear conservative dichroism could be derived theoretically. In the case of oscillatory flow rheological and rheo-optical properties could be compared directly. The present results provide evidence for the existence of a stress-optical law involving dichroism rather than birefringence. It differs from the stress-optical law for polymers but is similar to the case of concentrated suspensions of Brownian hard spheres. Conservative
Stress-Optical Relations in Immiscible Polymer Blends
dichroism is caused by scattering, which suggests that dichroism should not only depend on shape but also on size. Over a limited range of relevant sizes, around 1 micrometer, the size effect is however small. This explains the experimentally found relation between stresses and linear dichroism. A procedure is proposed to extend the stress-optical relation to smaller sizes by using a calculated correction factor for size.
Langmuir, Vol. 16, No. 8, 2000 3747
Acknowledgment. The authors are indebted to financial support for this project provided by the Geconcerteerde Onderzoeks Acties (GOA) of the K.U. Leuven. P.V.P. is a Postdoctoral Fellow of the Fund for Scientific Research-Flanders (FWO). We thank Dr. Jan Vermant for stimulating discussions. LA991406W
