7446
Langmuir 1999, 15, 7446-7453
Rheo-Optic Measurements on Dilute Suspensions of Hematite Rods Samuel J. Gason,‡ David V. Boger,‡ and Dave E. Dunstan*,† Cooperative Research Centre for Industrial Plant Biopolymers and Department of Chemical Engineering, The University of Melbourne, Parkville, Victoria 3052, Australia Received October 1, 1998. In Final Form: June 4, 1999 The microstructural changes induced by shear in dilute suspensions of colloidal Fe2O3 rods (aspect ratio ) 8.4) have been studied using a novel rheo-optic technique. The large extinction coefficient for the hematite system at the wavelength used (λ ) 392 nm) enables rheo-optic experiments on truly dilute suspensions to be undertaken. The elimination of any particle-particle interactions ensures that the measurements obtained can be directly related to the induced physical changes of the isolated particles. Optical absorbance spectra have been measured over a range of shear rates and angle of light polarization relative to the shear direction. The measured absorbance changes can be directly attributed to the shear-induced anisotropy in the suspension due to particle alignment. The results indicate that the hematite particles align with the direction of shear. The transition in the particle orientation is observed from a random Browniandominated system at Pe < 3, to a more ordered system under shear at Pe > 60. Plateau values in the measured spectra are observed at high Peclet numbers, indicating that a limiting degree of particle orientation occurs. Convergence of the observed extinction coefficients to zero at a polarization angle of ∼55° is seen at high Peclet numbers.
Introduction Colloidal rodlike particle suspensions such as hematite (Fe2O3) are of interest from both an industrial and fundamental point of view. Commonly used in the coating of polymeric substrates, such as in digital tape manufacture, an understanding of the suspension’s microstructural properties under shear is of paramount importance in controlling the properties of the final product. Rodlike particles under shear also provide a system whereby the effect of Brownian and hydrodynamic motions on a particle’s orientational order can be investigated at a fundamental level. To this end, an investigation into the effect of shear forces on the particle’s orientation has been undertaken. The use of rheo-optic methods is yielding valuable information regarding the physiochemical changes induced by shear.1-5 An historical and self-evident problem with classical rheology as a macroscopic measurement technique is that Newtonian and complex flow behavior are simply measured. The complex microstructural changes induced during shear thinning and elastic response may be inferred, but not singularly determined from macroscopic rheological measurement. In both polymer solutions and particulate suspensions, rheooptical studies have proven to be a popular technique for gaining further fundamental insight into the areas of fluid mechanics and physics at the molecular level.6-8 * Correspondence should be addressed to: d.dunstan@ chemeng.unimelb.edu.au. † Cooperative Research Centre for Industrial Plant Biopolymers. ‡ Department of Chemical Engineering. (1) Fuller, G. G. Annu. Rev. Fluid Mech. 1990, 22, 387-417. (2) Fuller, G. G. Optical Rheometry of Complex Fluids; Oxford University Press: New York, 1995. (3) Peterlin, A. Annu. Rev. Fluid Mech. 1976, 8, 35-55. (4) van de Ven, T. G. M. In Scientific Methods for the Study of Polymer Colloids and Their Applications; Canadau, F., Ottewill, R. H., Eds.; Kluwer Academic: 1990; pp 247-267. (5) Wagner, N. J. Curr. Opin. Colloid Interface Sci. 1998, 3, 391400. (6) Cottrell, F. R.; Merrill, E. W.; Smith, K. A. J. Polym. Sci. Part A: Polym. Chem. 1969, 7, 1415-1434.
The combination of rheological and optical measurement techniques enables the microstructural information of a system under shear to be characterized. The ability to predict the flow behavior of rodlike particle suspensions combined with verification of the significant body of theory (e.g., convective-diffusion equation9) should result from well-defined rheo-optic measurements.1,2 The most common methods used for rheo-optical measurements on particulate suspensions include light scattering,4 dichroism techniques,1 flow birefringence,7 and turbidity measurements.10,11 A beam of light is passed through the sample under shear and the response of the individual particles is characterized by the induced anisotropy of the transmitted light. The ordering of the microstructure is then deduced from the measured scattering or changes in intensity and polarization of the transmitted light.12 A rheo-optic technique has been developed that uses optical absorption spectrophotometry. The colloidal rod (Fe2O3) system, which is of both commercial and fundamental interest, has been examined. The large extinction coefficient of the hematite rods at 392 nm allows for the use of the absorption dichroism technique. Whereas in the past problems had been encountered because of the low scattering intensity of the particles with the light sources used, this technique allows for rheo-optic measurements to be collected for truly dilute systems. Results for the degree of orientation as a function of Peclet number are presented and show the transition from a random Brownian motion-dominated suspension regime to the alignment of the microstructure by the applied shear force. (7) Champion, J. V.; Gate, L. F.; Meeten, G. H.; Wood, P. R. Colloids Surf. A 1996, 111, 223-229. (8) Bender, J. W.; Wagner, N. J. J. Colloid Interface Sci. 1995, 172, 171-184. (9) Hinch, E. J.; Leal, L. G. J. Fluid Mech. 1972, 52, 683-712. (10) Anczurowski, E.; Mason, S. G. J. Colloid Interface Sci. 1967, 23, 522-532. (11) Anczurowski, E.; Mason, S. G. J. Colloid Interface Sci. 1967, 23, 533-546. (12) Johnson, S. J.; Frattini, P. L.; Fuller, G. G. J. Colloid Interface Sci. 1985, 104, 440-455.
10.1021/la981365o CCC: $18.00 © 1999 American Chemical Society Published on Web 09/03/1999
Rheo-Optics of Dilute Suspensions of Hematite Rods
Langmuir, Vol. 15, No. 22, 1999 7447
Theory for a Suspended Particle. The rotary Brownian motion of the particles in the quiescent state is represented by the rotary diffusion coefficient, Dr, which describes the time scale of the particle rotation. The combination of the thermodynamic properties, particle dimensions, and solvent viscosity led to the formulation of the rotary diffusion coefficient of the particle perpendicular to its symmetry axis. The calculation for Dr, shown below, highlights its dependence on particle aspect ratio and solvent properties.
Dr ) kT/6VpηsrK
(1)
where k is Boltzmann’s constant, T is the temperature, and ηs is the solvent viscosity. The particle’s volume, Vp, is calculated as below
Vp )
4π 2 ab 3
(2)
where 2a is the major axis length and 2b is the minor axis length. The material constant, rK, is calculated for the blunt-ended ellipsoid using the aspect ratio, rp ) a/b13
rp2
r-1)3], the steady-state orientation distribution remains independent of the initial particle orientation. Rotary Brownian motion has the effect of randomizing the orientation of the particles of the suspension in the quiescent state. The final distribution of the particle orientation is then observed as a compromise between the undisturbed Jeffery orbits and the random Brownian motion of the particles. The particle’s motion, while being dominated by the hydrodynamic motion, has an inertial component described by the aspect ratio term. However, theoretical predictions for the steady-state behavior of a particle showing Jeffrey’s orbitals and no Brownian motion [Pe > (r + r-1)3] show that the final particle orientation is dependent on the initial condition.9 Observation of a freely suspended particle’s response to shear has led to many rheo-optical studies. Fuller1,2 and Peterlin3 provide reviews of birefringence and dichroism techniques for the measurement of the suspension’s microstructural behavior under shear. From Jeffery, the orientation of a noncolloidal spheroid with respect to the rotational symmetry axis (2a, major axis) and the perpendicular axis (2b, minor axis) is specified in polar coordinate angles θ and φ to the main axis of rotation.
(3)
θ˙ )
(bγ4˘ ) sin 2θ sin 2φ
(5)
The Peclet number, Pe, is a dimensionless number relating the shear forces to the Brownian thermal “forces” acting on the particle in solution.13 In using the Peclet number the relation between the particle’s response to shear and the randomizing effect of Brownian motion can be understood. The dominant forces relate to the magnitude of the Peclet number. The Peclet number is defined as
φ˙ )
(2γ˘ )(1 + b cos 2φ)
(6)
r
K)
3(ln 2rp - 0.5)
γ˘ Pe ) Dr
(4)
where γ˘ is the shear rate and Dr the rotary diffusion coefficient. Initial studies by Jeffery into the dynamics of suspensions focused on the limiting case for Pef∞. The motion of the particle is dominated by the applied shear force, with the particles following a periodic vorticity orbital, without the diffusive effect of rotational Brownian motion.14 The orientation of the particle was subsequently calculated from the solving of the equations of motion for the spheroid suspended in a Newtonian fluid and showed the particle to transverse an infinite number of closed orbits. Further to this, Mason and Anczurowski derived the orientation function for a collection of long particles of high aspect ratio and no Brownian motion, initially of random orientation in the quiescent state under shear.10,11 The turbidity experiments of Mason were conducted on large fibers suspended in a viscous solvent and by virtue of the particle size the Drf0 and Pef∞. Orientation and subsequent relaxation is attributed to the measurement of the particle’s Jeffery orbitals. As the diffusive motion of the particles is ignored for the equations of orientation, they are only valid for large values of Pe. The convective-diffusion equations for particle motion describe three regimes for the state of the particle.9 For a particle where Pef0, the result for the probability function is well-known. For a particle of weak Brownian motion in the intermediate regime [Pe > 1, Pe < (r + (13) Brenner, H. Int. J. Multiphase Flow 1974, 1, 195-341. (14) Jeffery, G. B. Proc. R. Soc. London Ser. A 1922, 102, 161-179.
b)
(rp2 - 1)
(7)
(rp2 + 1)
where b is the anisometry parameter. Combination of the particle distribution function, Φ(θ,φ), and the optical anisotropy of the particle leads to the birefringence.
∆n ) nI - nII ) (2πc/Fηs)(g1 - g2)f(β,b)
(8)
cot 2χ ) h(β,b)
(9)
where
f(β,b) ) χ)
[
( (
) ] ) ]
β2 6b2 βb 11+ + ... 15 72 35
[
β β2 24b2 π 11+ + ... 4 12 108 35
(10) (11)
and β is the Peclet number. Extinction of Light. The extinction of light is defined as the attenuation of the electromagnetic wave by scattering and absorption as it transverses a particular medium.15 If multiple scattering is negligible and the irradiance of the light beam is exponentially attenuated from Ii to It when transversing a medium of distance l, then
Ii ) exp(-Rl) It
(12)
where (15) Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic Press: New York, 1969.
7448
Langmuir, Vol. 15, No. 22, 1999
R)
4π λ
Gason et al.
(13)
and
Cext ) Cabs + Csca
(14)
where is the index of attenuation or the extinction coefficient and λ is the wavelength of the light. Accordingly, the total extinction cross-section, Cext, for the particle comprises the energy abstracted from the incident beam by both absorption and scattering. Cabs and Csca are the absorbance and scattering cross-sections. The absorbance of light by a prolate spheroid is directly proportional to the length scale or aspect ratio of the particle. Therefore the ratio of the absorbance in the direction of the major axis to the absorbance in the direction of the minor axis will be 1:rp, which for the sample used in this study is 1:8.4. However, the scattering due to the prolate spheroid is proportional to the aspect ratio by 1:rp6.15 This result is obtained from the Rayleigh scattering function for the intensity of scattered light. Whereby the intensity of scattered light for a spherical particle is proportional to the radius of the particle, this proportionality includes the aspect ratio for the prolate spheroid.15 Any scattering of light by the particle in the direction of the major axis will therefore dominate over any scattering of light in the minor axis direction. For the obstruction of light by a conductive medium the complex refractive index is defined as:
m ) n(1 - i)
(15)
The real part, n, arises from the phase shift and determines the phase velocity of the incident beam while the imaginary part, n, is the damping factor, which arises from the attenuation (absorption and scattering) of the light as it propagates through the medium. Materials and Methods A sample of hematite rods (Fe2O3) with a major axis (2a) length of 320 nm, minor axis (2b) length of 38 nm and aspect ratio, rp, of 8.4 was supplied the 3M Corporation (St. Paul, MN). A transmission electron micrograph (TEM) of the particles was used to confirm their dimensions and spindle geometry (Figure 1).16 The rods were dialyzed for 1 month with a triple-distilled water solution of pH 3 (HCl) and an ionic concentration of [KCl] ) 10-3 M. The particles were dispersed by sonication in tripledistilled water/glycerol solutions at pH 3 and [KCl] ) 10-3 M. The addition of the background electrolyte enables the control of the electric double layer (and Debye length, κ-1) surrounding the particles. The Debye length of the system can by calculated using
κ-1 )
(
8πe2NaI 1000dckT
)
-1/2
(16)
where e is the elementary electric charge, dc is the dielectric constant, Na, k, and T have their usual meanings, and I is the ionic strength of the solution
∑z M 2 i
I ) 1/2
i
(17)
i
The effect of glycerol on the size of the double layer is shown in Table 1, where the reduction in the dielectric constant is seen to slightly reduce the size of the double layer. (16) Solomon, M. J.; Boger, D. V. J. Rheol. 1998, 42, 929-949.
Figure 1. Transmission electron micrograph of hematite particles. Magnification (A) × 15 000 and (B) × 73 000, with scale shown. Particles are of spindle geometry with an aspect ratio of 8.4. Table 1. Double Layer Length for 10-3 M KCl in Glycerol/Water Mixtures conc. glycerol (%w/w)
dc
κ-1 (Å)
0 60 80 90 95 100
78.5 56.9 49.7 46.1 44.3 42.5
96.2 81.9 76.5 73.7 72.2 70.7
The addition of hematite to the solution reduced the viscosity of the suspension, as shown in Figure 2. The reduction in viscosity observed in preparation of the suspensions was attributed to residual water in the hematite sample. A sample of hematite suspension was dried to a constant mass and was shown to contain 4.3% water by weight. All flow curves were measured using the Carri Med CSL2100 controlled stress rheometer with the cone and plate geometry. The suspensions display Newtonian behavior over the shear rate range of 0 to 1400 s-1. The ultraviolet/visible (UV/vis) absorption spectra, using a Cary 3E spectrophotometer, collected between 190 and 900 nm show the absorption maxima, λmax, at 392 nm, as shown in Figure 3. All subsequent rheo-optic experiments were performed with the incident light beam at λmax ) 392 nm. For the UV/vis absorption experiments the Beer-Lambert law is used to relate the observed absorbance to the cross-sectional of the particles by the extinction coefficient, .
Rheo-Optics of Dilute Suspensions of Hematite Rods
Langmuir, Vol. 15, No. 22, 1999 7449
Figure 2. Viscosity of the glycerol/water mixtures, with and without Fe2O3. Reduction in suspension viscosity due to water in hematite sample.
Figure 4. Diagram of the Couette geometry used, showing the path of the light beam and the angle of polarization. The light beam passes through the sample perpendicular to the flow direction. concentration was calculated for the particle having dimensions of 338 nm × 56 nm and a density of 2.06 g/mL and was calculated to be φ ) 0.115. Experimentally, the volume fraction including the double layer was φ ) 0.0015 for 1.57 g/L hematite in pure glycerol. This number is slightly reduced by increasing the water concentration. The spectroscopic technique used a quartz parallel cylinder, as shown in Figure 4. The Couette geometry is used to create a shear flow field, which also allows spectroscopic measurements to be carried out. The ratio of the radii of the two quartz cylinders,
κ)
Figure 3. UV/vis spectrum of hematite rods in water, showing λmax of 392 nm.
Abs ) cl
(18)
where Abs is the measured absorbance, c is the particle concentration, and l the path length. As the concentration and path length remain constant throughout the experiment, any changes to the absorbance can be attributed to changes in the extinction coefficient or cross-sectional area of the particle. These systems remained stable with no sign of flocculation or settling over time. The overlap concentration was calculated using16,17
na3 ) 1
(17) Berry, D. H.; Russel, W. B. J. Fluid Mech. 1987, 180, 475-494.
(20)
ensures that a near constant shear rate is maintained across the gap. The inside cylinder (R1 ) 9.50 mm) rotates at controlled speeds and the outside cylinder (R2 ) 10.00 mm) is fixed. The Couette cell is placed in the Cary 3E spectrophotometer and manually adjusted so the flow field is perpendicular to the incident beam. This is achieved by manually adjusting the cell position laterally until a minimum in the absorbance is seen. Spectra were then collected with shear rates varying from 0 to 160 s-1. Calculations of the critical Reynolds number, (Rec), and critical Taylor number, (Tc), show the flow system to be laminar and free of recirculating vortices (Taylor vortices18).
γ˘ )
(19)
to be a volume fraction of φ ) 0.06. The working concentration was chosen to be φ ) 0.0006 or 1.57 g/L, eliminating any particleparticle interactions in a truly dilute suspension. The addition of salt formed the double layer about the particle that would induce some effect on the particle’s rotation. The overlap
R1 > 0.95 R2
Rec )
(
-2Ω R1 1R2
)
Ω1κR21F ηs
()
trans
2
)
41.3 (1 - κ)3/2
(21)
(22)
where γ ) shear rate (s-1) (18) Taylor, G. I. Proc. R. S. London Ser. A 1923, 105, 541-542.
7450
Langmuir, Vol. 15, No. 22, 1999 Ω1 ) angular velocity (rad s-1) R1, R2 ) inner and outer radii (m) κ ) (R1/R2) F ) solvent density (kg m-3) ηs ) solvent viscosity (kg m-1 s-1)
An important extension of the above experiments involves using polarized light whereby the anisotropy of the flow system can be observed. The incident beam is polarized by using a GlanTaylor polarizer (CARY, mounted with vernier and dial, extinction ratio 1:10-5) orientated between ψ ) 0° (parallel) and ψ ) 90° (perpendicular) to the flow field, allowing angular-dependent extinction changes to be recorded, as shown in Figure 4. Baseline spectra for the absorbance due to the polarizer were recorded, enabling the extinction coefficient of the polarizer to be subtracted in subsequent experiments. The Couette cell was washed with concentrated acid solution and then rinsed with a pH 3/[KCl] ) 10-3 M solution before each experiment to ensure the dissolution of any hematite adsorbed to the cell. The suspension is carefully placed in the cell, ensuring no air bubbles formed, and then allowed to stand undisturbed for 15 min. The particle orientation of the suspension is then assumed to be completely random (isotropic) in the quiescent state. The shear cell was loaded into the spectrophotometer with the light beam wavelength set to 392 nm, the absorbance maximum for hematite. A shear rate, between 0 and 160 s-1, was set for 30 s with the absorbance measured over this time. The cessation of shear was followed by 2 min at zero shear, during which time the absorbance was observed to return to a constant value, indicating that the suspension had returned to the isotropic state. A slight decay in the absorbance was observed immediately after the cessation of shear; this is highlighted in the cases of higher glycerol concentration and is explained by the particles returning to the random isotropic state. This procedure was followed for higher shear rates (up to 160 s-1).
Results and Discussion Dilute Suspensions. The rheology of the Fe2O3 suspension in glycerol/water shows Newtonian behavior between the shear rates of 0 to 1400 s-1. Stability against sedimentation and flocculation was observed over a period of 1 month (the duration of the experiments). The Peclet number is the ratio of the viscous to thermodynamic forces acting on the particle in solution, as defined earlier (eq 4). The value for the rotary diffusion coefficient, Dr, is solvent viscosity dependent and could therefore be controlled and reduced by the addition of glycerol to the system. In this way glycerol may be used to control the Brownian motion of particles. Table 2 shows the value for Dr in solutions of varying concentration of glycerol used for the hematite rods of aspect ratio of 8.4. Experiments determining the extinction coefficient were conducted using cuvette path lengths of 1 mm and 10 mm over a range of particle concentrations. The values obtained for from the cuvette cells are in good agreement with that calculated using the shear cell. The extinction coefficient for the hematite rods at 392 nm was calculated from experiments to be 1.91 m2 g-1. Calculation of the extinction coefficient in the cuvette also confirms the gap width of the Couette cell to be 0.5 mm. The axis lengths were used to calculate the theoretical cross-sectional area of the particle and compared with the calculations, from the extinction coefficient, for the experimental cross-sectional area. The simple theoretical particle area calculations included the end-on area (1.13 × 10-15 m2) as well as the side-on area (1.22 × 10-14 m2) and the maximum particle cross-sectional area (8.04 ×
Gason et al. Table 2. Calculated Rotary Diffusion Coefficient, Dr, for Hematite in Glycerol/Water Systems glycerol (w/w%)
viscosity (Pa s)
Dr (s-1)
0 60 80 90 96 97.5 100
0.001 0.01524 0.05766 0.2081 0.656 0.7256 1.206
275 17.857 4.7633 1.3215 0.4192 0.3836 0.2280
10-14 m2). From the value for the maximum particle crosssection, the ensemble average projected area (calculated projected area) is calculated as projected area ) 2/3 maximum area, equaling 5.3 × 10-14 m2 . The experimental particle cross-section area, which predicts the area swept out by the freely suspended particle randomly rotating, was calculated from the measured extinction coefficient. The density of the particle used was 4.72 (g/mL). The extinction coefficient, which predicts the cross-sectional area for 1 g of substrate, was reduced to the particulate level, yielding 2.18 × 10-15 m2 as the projected area per particle. The experimental particle area was found to be much less than that for the calculated projected area. The extinction of light arises from the combination of the processes of absorption (i.e., the transformation of the electromagnetic energy into other forms) and scattering by the particles. The direct association of the absorbance to the particles’ ensemble average cross-sectional area requires further examination. At λmax ) 392 nm the refractive index for hematite is 1.95 + 0.88i.19 The relatively large imaginary component of the refractive index indicates that both absorbance and scattering play important roles in the extinction of light by hematite, as shown in Figure 3. Normalization of the spectra reveals that the extinction cross-section of light can be attributed to both scattering and absorbance. The scattering of the particles characterized by the λmax ≈ 200 nm and the absorbance of the light at 392 nm combine to give the overall spectra shown. At 392 nm the ratio of the contribution of each phenomenon to the peak is shown to be 2:1. The scattering and absorption phenomena and quantum efficiency of hematite may explain the anomalies observed in the calculation of the particle dimensions from the experimental extinction coefficient. The extinction efficiency of the hematite used in these experiments can be used to explain this result and is defined as the number of photons being absorbed by the sample to the number of photons hitting the particle.20 The absorption process is reliant on the orientation of the absorbing species and the probability of it intercepting a photon, as well as there being an electron in a favorable state for absorption to occur. The extinction efficiency of the hematite sample, calculated as the ratio of measured particle extinction coefficient to calculated particle extinction coefficient, was found to be 0.042. Rheo-Optic Measurements. The rheo-optic measurements were conducted as discussed earlier and the spectra were collected in the form shown by Figure 5. Increasing the solvent viscosity, as previously mentioned, extended these experiments to include a large range of Pe. The change in absorbance was calculated by subtracting the average of the preshear and postshear absorbance from the shear-induced absorbance. The observed drift in the absorbance baseline is believed to be due to microvariations in the cell position upon shear. This drift was (19) Welch, R. M.; Cox, S. K. Appl. Opt. 1978, 17, 3159-3168. (20) Hsu, W. P.; Matijevic, E. Appl. Opt. 1985, 24, 1623-1630.
Rheo-Optics of Dilute Suspensions of Hematite Rods
Langmuir, Vol. 15, No. 22, 1999 7451
Figure 7. Change in extinction coefficient versus applied Peclet number for 1.57 g/mL hematite in glycerol/water, highlighting limiting plateau value. Unpolarized spectra.
Figure 5. Absorbance (at λmax) versus time for 80% glycerol/ water. Each peak corresponds to an applied shear rate between 0 and 100 s-1. Unpolarized spectra.
Figure 8. Change in extinction coefficient versus applied Peclet number for 1.57 g/mL hematite in glycerol/water, highlighting transition from thermodynamic system to hydrodynamic system.
Figure 6. Change in extinction coefficient versus applied shear rate for 1.57 g/mL hematite in 80% glycerol/water. Unpolarized spectra.
eliminated in subsequent experiments by modification of the cell. The change in absorbance value was then converted to the change in extinction coefficient. By plotting change in extinction coefficient versus shear rate it can be shown that the extinction coefficient is both shear rate and solvent viscosity dependent. This observation is highlighted in Figure 6. The extinction of light by the suspension arises from the absorbance and scattering at a specific wavelength by the individual hematite rods in a two-dimensional plane. The extinction cross-section is taken from the ensemble average, which is composed of particles in all orientations in zero shear. The attenuation of the light is then attributed to the increased cross-sectional area of the particles perpendicular to the light beam due to their orientation in the shear. The observed changes in the absorbance spectra are due to shear-induced alterations
in the ensemble average cross-section of the particles relative to the light beam. Therefore, scaling the shear rate with the Peclet number enables the changes in extinction to be related to changes in the average orientation of the hematite rods. Figures 7 and 8 show the change in extinction coefficient versus Peclet number. As expected, there is no observed change in the extinction coefficient for Pe < 1, where the random Brownian or thermal forces dominate the applied shear. As the Peclet number is increased above 3, the imposed shear forces become important, leading to anisotropy in the suspension on the microstructural level. The appearance of a limiting value in the change in extinction coefficient with an increase in the Peclet number indicates a finite orientation of the particles (Figure 7). The transition from a Brownian-dominated system at Pe < 3 to a system where the particle motion is dominated by hydrodynamics (at Pe > 60) is shown in Figure 8. The definition of the hydrodynamic regimes as provided by Hinch and Leal9 can be used to describe the particle motion and the controlling physics. At the low Pe values (Pe < 1) the particle orientation can be described as random because of Brownian motion. It can be shown that these measurements were conducted in the intermediate region where inertially induced diffusion must be considered even though the particle motion is primarily governed by the applied shear force. The expression describing strong shear [Jeffery regime, Pe . (r + r-1)3] does not hold for this case
7452
Langmuir, Vol. 15, No. 22, 1999
Gason et al.
Figure 11. Change in extinction coefficient versus Peclet number for polarized light. Positive 4 for light (θ ) 0°) and negative 4 for light (θ ) 90°).
Figure 9. Absorbance for light polarized parallel (θ ) 0°) to the flow direction (at λmax) versus time for 80% glycerol/water. Each peak corresponds to an applied shear rate. (Note that the difference in magnitude of absorbance from the unpolarized spectra is due to the absorbance of the polarizer.)
Figure 12. The dichroic ratio, R, for the hematite in glycerol at a range of Peclet numbers.
Figure 10. Absorbance for light polarized perpendicular (θ ) 0°) to the flow direction (at λmax) versus time for 80% glycerol/ water. Each peak corresponds to an applied shear rate.
where the highest applied Pe is ≈ 420, and (r + r-1)3 ≈ 600. The calculation for the particle orientation distribution function therefore becomes a nontrivial task. A theoretical treatment will be provided in forthcoming work. Changes in the extinction spectra are explained by particle alignment relative to the light beam. In the quiescent state the random array of rods project a certain cross-sectional area perpendicular to the beam. The application of shear, where Pe > 1, aligns the rods in the direction of shear, increasing the ensemble average projected area parallel to the light beam. Polarization Experiments. The use of polarized light as the incident beam enabled further insight into the alignment of the rod systems under shear to be gained.
Figure 13. Change in extinction coefficient versus Peclet number for light polarized at different angles to the shear direction. 90% glycerol.
Experiments were conducted where the angle of polarized light was varied between ψ ) 0° (parallel) and ψ ) 90° (perpendicular) to the direction of shear. Figures 9 and 10 show spectra collected for the polarized light. The extinction coefficient increases with shear rate when the incident light is polarized parallel to the shear direction and decreases when the incident light is perpendicularly polarized. (See Figure 11.) When plotted as the dichroic
Rheo-Optics of Dilute Suspensions of Hematite Rods
Langmuir, Vol. 15, No. 22, 1999 7453
52-58°. The random state is the same as projecting the rods at an angle of ∼55° onto the absorbing plane. For light that was polarized parallel to the direction of flow, the increase in extinction coefficient is shown to be dependent on the Peclet number. By the same rationale as discussed previously, this increase is attributed to the ensemble of particles aligning with the direction of shear. Therefore an increase in cross-sectional area is observed in the flow direction. The induced anisotropy of the individual particles aligning with the flow also leads to the decrease in the extinction coefficient seen for the perpendicularly polarized light as seen in Figure 11.
Figure 14. Change in extinction coefficient versus Peclet number for light polarized at different angles to the shear direction. 96% glycerol.
Figure 15. Change in extinction coefficient versus degree of polarization to the direction of shear.
ratio, R ) ||/⊥|, the results are shown to converge to a constant value of approximately 3, as shown in Figure 12. Further polarization experiments were conducted to find the limiting angle of orientation. By increasing the degree of polarization with increasing Peclet number, the ∆ was seen to decrease to a negative value, passing through a point of zero change. Here negative ∆ values indicate that the absorbance is less than the absorbance values for the quiescent suspension. Figures 13 and 14 show the effect of polarization of the incident light on the resultant ∆ for hematite in 90% glycerol and 96% glycerol. By selecting Peclet numbers (40, 80, 125) common to both sets of results, an approximation of the limiting value of orientation can be gleaned. From Figure 15, where the ∆ is plotted against degree of orientation, the apparent limiting value of particle orientation is in the range ψ )
Conclusions Rheo-optical experiments have been conducted on suspensions of hematite rods to quantify the relation between the Peclet number and flow-induced anisotropy of the suspension. Normalization of the data led to the formulation of a master curve (see Figure 8) for the measured spectral changes with Peclet number. By measuring the extinction coefficient it is possible to relate the spectral changes, at a given shear rate, to the orientation of the rods. The extinction coefficient was seen to increase for unpolarized light with increasing Pe > 1, indicating an increase in the cross-sectional area seen by the light beam. An increase in the extinction coefficient was also observed for light polarized parallel (ψ ) 0°) to the flow. Light polarized perpendicular (ψ ) 90°) to the direction of flow showed a decrease in extinction coefficient with Pe > 1. An apparent limiting degree of particle orientation was found to be ψ ) 52-58° for the highest Peclet number. The limiting value seen in the change in extinction coefficient is indicative of a finite value of particle orientation. The experimental observations can be explained by the transition between the randomizing effect of the thermodynamic Brownian motion of the particles and the applied hydrodynamic forces. Future work will compare the theoretical particle orientation distribution function with the experimental dichroic observations. Acknowledgment. S.G. gratefully acknowledges the assistance of the Commonwealth Australian Postgraduate Award Scholarship and the Australian Research Council Special Investigators Grant to the A.R.C. held by Professor D. V. Boger. Part of this work is supported by the Advanced Minerals Product Research Centre, a Special Research Centre funded by the Australian Research Council and the Cooperative Research Centre for Industrial Plant Biopolymers. The useful discussions with Prof. Derek Chan and Dr. Michael Solomon and the preparation and measurement of the hematite and TEM by Mr. Thearith Ung are gratefully acknowledged. The work of Mr. Herbert Groiss in machining the precision quartz Couette cell is also acknowledged. LA981365O
