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Rheology and Microstructure of Mixtures of Colloidal Particles William J. Hunt and Charles F. Zukoski* Department of Chemical Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 Received April 26, 1996. In Final Form: September 18, 1996X The microstructure and flow properties of bimodal mixtures of charge-stabilized colloidal particles are investigated. Using small-angle neutron scattering (SANS), an initially ordered lattice of monodisperse 220 nm diameter particles at a volume fraction near the order/disorder phase transformation boundary is observed to maintain long range orientational order with the addition of a small number density of 54 nm diameter particles. As the number density of these “defects” is increased, the colloidal crystal loses long range order abruptly. Changes in the rheological properties of these suspensions are correlated with changes in microstructure. The initally ordered lattice is a viscoelastic solid, but with the addition of a sufficient number density of “defects”, the suspension behaves as a purely viscous fluid. The changes in rheology and microstructure are not observed by simple dilution of the 220 nm particles.
1. Introduction In this work we explore the microstructural and rheological consequences of adding small (54 nm diameter) charge-stabilized particles to a suspension of large (220 nm diameter) charge-stabilized particles. The suspension of large particles is at a volume fraction just above the order/disorder phase transition boundary, and small particles are added such that, at their highest density, there are two large particles for every small particle in the suspension. However, the volume fraction of the small particles is always less than 0.002 while the volume fraction of the large particles is effectively constant at 0.28. These studies were undertaken to determine the density of small particles or defects that the crystals of large particles can sustain prior to disordering. Investigations of the microstructure and mechanics of bimodal mixtures containing Brownian particles are limited.1-6 In studies of suspensions containing effective hard sphere particles a minimum in viscosity is often reported when the number density of the small and large particles is varied at constant total volume fraction. This behavior is attributed to the more efficient utilization of available volume in the mixture when compared to the monodisperse suspensions.7,8 In these systems, little is known about viscometric properties other than a viscosity at a given shear rate. In separate studies, both modeling9,10 and experimental,11 surface ordering and bulk phase separation of bimodal suspensions have been reported. On the other hand, on the basis of a linear variation in elasticity as large and small particles are mixed at fixed total volume X Abstract published in Advance ACS Abstracts, December 1, 1996.
(1) Pusey, P. N.; van Megen, W.; Barlett, P.; Ackerson, B. J.; Rarity, J. G.; Underwood, S. M. Phys. Rev. Lett. 1989, 63, 2753. (2) Hoffman, R. L. J. Rheol. 1992, 36, 947. (3) Chaffey, C. E.; Wagstaff, I. J. Colloid Interface Sci. 1977, 59, 63. (4) Storms, R. F.; Ramarao, B. V.; Weiland, R. H. Powder Tech. 1990, 63, 247. (5) Poslinski, A. J.; Ryan, M. E.; Bupta, R. K.; Seshadri, S. G.; Frechette, F. J. J. Rheol. 1988, 32, 751. (6) Farris, R. J. Trans. Soc. Rheol. 1968, 12, 281. (7) Yu, A.-B.; Zou, R.-P.; Standish, N. J. Am. Ceram. Soc. 1992, 75, 2765. (8) McGeary, R. K. J. Am. Ceram. Soc. 1962, 44, 513. (9) Kranendonk, W. G. T.; Frenkel, D. Mol. Phys. 1991, 72, 679. (10) Zeng, X. C.; Oxtoby, D. W.; Rosenfeld, Y. Phys. Rev. A 1991, 43, 2064. (11) Kaplan, P. D.; Rouke, J. L.; Yodh, A. G.; Pine, D. J. Phys. Rev. Lett. 1994, 72, 582.
S0743-7463(96)00416-7 CCC: $12.00
fraction, Lindsay and Chaikin12 suggest a “colloidal glass” is formed during the mixing process. Clearly microstructural and rheological properties of bimodal mixtures offer a diverse array of phenomena. A full mapping of this behavior is difficult due to the number of parameters which can be varied. Rather than attempt such an extensive study, we report on the mechanical consequences of the disordering of a colloidal crystal by the introduction of “defects”. 2. Experimental Section Two polystyrene (PS) latex suspensions were used in this work. The first were 220 nm in diameter, termed large or big particles for the remainder of the text, prepared by the emulsifierfree, emulsion polymerization techniques of Juang and Krieger.13 The 54 nm or small particles were prepared by emulsion polymerization described by Hawkett et al.14 After synthesis, the latex was filtered through glass wool to remove any large pieces of flocculated material and cleaned by the procedure of Homola and James.15 The resuspended latices were placed in SpectraPor 4 dialysis tubing (molecular weight cutoff 14 000) and dialyzed against deionized water. The dialyzate was changed regularly, and dialysis was considered complete when the conductivity of the dialyzate dropped below 10 µS/cm. The latex was then dialyzed against the desired ionic concentration for 2 weeks with regular changing of the dialyzate. Once the correct ionic strength was reached, as determined by monitoring the conductivity of the dialyzate, all solutions into which the latices come in contact had that ionic strength. The latex was concentrated by immersing the dialysis tubes into a polyethylene glycol (molecular weight 20 000, Union Carbide) solution of roughly 1 wt % in deionized water. The difference in osmotic pressures between the solutions causes water to flow from the latex to the polymer solution. When the volume fraction of the large and small particles had reached roughly 0.30 and 0.07, respectively, the dialysis tubes were removed from the polymer solution and the contents of the tubes combined in separate vessels for use as stocks. The large and small stock volume fractions were determined to be 0.283 and 0.063, repsectively, by weight loss on drying assuming a polymer density of 1.05 g/cc.16 Particle size distributions were examined by transmission electron microscopy (TEM) and dymanic light scattering (DLS). The large particles were determined by both techniques to be (12) Lindsay, H. M.; Chaikin, P. M. J. Chem. Phys. 1982, 76, 3774. (13) Juang, M. S.-D.; Krieger, I. M. J. Polym. Sci. 1976, 14, 2089. (14) Hawkett, B. S.; Napper, D. H.; Gilbert, R. G. J. Chem. Soc., Faraday Trans. 1 1980, 76, 1323. (15) Homola, A.; James, R. O. J. Colloid Interface Sci. 1977, 59, 123. (16) Hachisu, S.; Kobayashi, Y. J. Colloid Interface Sci. 1974, 46, 470.
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220 ( 4 nm in diameter. By dynamic light scattering, the small particles were 54 ( 10 nm in diameter. TEM analysis shows the particles have a slightly bimodal distribution with diameters of 60 and 40 nm with a number density distribution of about 3:1. The DLS measurement of 54 nm is used as the size for any calculations for number density or number density ratio in the mixtures. 2.1. Neutron Scattering. The SANS experiments in this work were carried out at the Department of Commerce National Institute of Standard and Technology Cold Neutron Research Facility17 located in Gaithersburg, MD, using the shear cell designed by Straty et al.18 The intensity of the scattered neutrons is measured as a function of the scattered wave vector, q, defined by eq 1
q)
θ 2π r0 4π sin ≈ λ 2 λ D
(1)
Where λ is the neutron wavelength, θ is the angle between the scattered wave vector and the incident wave vector, r0 is the radial distance on the detector measured from the undeviated beam path, and D is the sample to detector distance. All SANS measurements were made in the velocity-vorticity plane (kvke) using a wavelength of 12 Å (14% fwhm). The sample to detector distance, D, used was 13.65 m. 2.2. Rheology. Rheological characterization of the mixtures was carried out on a Bohlin CS controlled stress rheometer with a 25 mm ID Couette device with a 1.25 mm gap. When higher applied shear stresses were required, a 14 mm ID Couette geometry with a 0.75 mm gap was used. Creep experiments were performed to determine the viscosity and the presence and magnitude of a dynamic yield stress. Attempts were made to measure the elastic modulus of the latices but were unsuccessful. The stock suspension of the large particle latex had an elastic modulus that was near the sensitivity limit of the insturment (approximately 2 Pa) and consequently had a large uncertainty, roughly equivalent to the measurement. A dynamic yield stress, τdy, will be defined as a shear stress above which continuous, steady state deformation will occur. For applied stresses below the dynamic yield stress, plastic deformation can occur. In the experiments carried out here, 15 min was allowed to elapse after applying the stress before the steady state rates of deformation were measured. If, after this time period, the shear rate, γ˘ , had not reached a steady state, the stress was taken to be below τdy. For stresses above the dynamic yield stress, a steady state rate of deformation is typically achieved in 1-2 min. All viscosity measurements were performed for 15 min of creep, after which the sample was allowed to sit undisturbed for 30 min before the next measurement was taken. The sample is in its equilibrium state after this period of time and shows no shear history effects. 2.3. Preparation of Mixtures. The particles have a negative surface charge defined by synthesis conditions, and they will repel each other when brought into proximity through doublelayer overlap. Consequently, there is no flocculation upon mixing the two stock latices. No flocculated material settling out of the mixture was observed over the period of several months. Number ratios are calculated by determining the stock latex number density from the particle diameter. A known weight of each latex is then added to a vessel, and the mixture is vigorously agitated by hand to mix the two stocks. All samples were prepared using normal water (H2O) as the solvent for both SANS and rheology studies. The large particles are drawn from a stock suspension at 28.3% solids by volume. The pure large particles display long range orientational order at volume fractions above 0.26. It is important to note that at no time during the mixture experiments does the overall volume fraction drop below this transition point. Even for the largest number density of small particles, the volume (17) Glinka, C. J.; Rowe, J. M.; LaRock, J. G. J. Appl. Crystallogr. 1986, 19, 427. (18) Straty, G. C.; Hanley, H. J. M.; Glinka, C. J. J. Stat. Phys. 1991, 62, 1015.
fraction of the large particles is 0.275 while the small particle volume fraction in this mixture is 0.002.
3. Results and Discussion 3.1. Microstructure of Mixtures. The mixtures were loaded into the quartz Couette device (1 mm gap) and sheared at 10 s-1 for 5 min. The shearing was stopped, and the samples were allowed to sit undisturbed for 5 min before equilibrium scattering patterns were taken. The reference structure used for comparision in looking for changes produced on addition of small particles is that of the pure large particles. The SANS investigation of the stock suspension of large particles demonstrated microstructural transitions consistent with previous work on monodisperse colloidal crystals.19-21 The large particles formed randomly stacked densely packed planes parallel to the walls of the cell. At equilibrium, this gives rise to six primary intensity maxima in the scattered neutron intensity (Figure 1). The long axis of the hexagon points along the direction of vorticity in the shear cell with a unit vector ke, indicating that the (111) direction of the crystal is oriented along the velocity direction with unit vector kv. As the sample is sheared, the scattered neutron intensity pattern progresses through a series of characteristic patterns. At 1 s-1, the primary hexagonal pattern of intensity maxima is replaced by a ring of uniform intensity at the same q values at which the intensity maxima were observed under equilibrium. This pattern has been observed in a number of studies and is associated with a polycrystalline state where the short range order is maintained but the long range orientation between particles is disrupted. The pattern changes to a four “spot pattern” at 10 s-1, where four intensity maxima are observed in the inner ring. This pattern is characteristic of a sliding layer or shearing layer structure where densely packed planes parallel to the cell wall slip over one another. The in-plane correlation of particle positions is time invariant; however, the correlation between similar points in the plane particles changes with time. In the sliding layer structure, each layer freely slips over the neighboring layers. In the shearing layer structure, crystals are strained to their elastic limit and strain is localized to zones or bands and the deformation rate is not uniform across the gap. The off-axis scattering experiments21 required to distinguish these structures were not carried out. As the shear rate is increased to 100 s-1, the scattered neutron intensities return to a ring pattern where no intensity maxima are observed and the SANS pattern is close to that of a fluid. This loss of orientational order is referred to as shear melting, and in high solids loading systems can be accompanied by the onset of shear thickening.22 The addition of the small particles does not affect the average spacing of the large particles. This is shown by the lack of shift of the maximum intensity of the scattered neutrons in Fourier space (i.e., the position of the ring in Figure 2 does not shift). The scattering in the range of Fourier space used here is dominated by the largest objects with the scattered intensity proportional to the volume of the scatterer squared. At the largest volume fraction of small particles, the volume fraction of the large particles is 0.275 while the small particles are at roughly 0.002. Consequently, of the neutrons that strike the detector, roughly 150 are scattered from the large (19) Chen, L. B. The Dynamic Properties of Concentrated Charge Stabilized Suspensions. Ph.D. Thesis, University of Illinois, 1994. (20) Clark, N. A.; Hurd, A. J.; Ackerson, B. J. Nature 1979, 281, 57. (21) Chow, M. K.; Zukoski, C. F. J. Rheol. 1995, 39, 33. (22) Chen, L. B.; Chow, M. K.; Ackerson, B. J.; Zukoski, C. F. Langmuir 1994, 10, 2817.
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Figure 1. Scattered neutron intensities in the kv-ke plane for pure large particles at various shear rates: (a) 0 s-1; (b) 1 s-1; (c) 10 s-1; (d) 100 s-1. Colors moving from blue to red to yellow to white indicate more neutron strikes. All patterns are normalized to 107 incident neutrons. The center of each image is covered by a beam stop.
particles for every one that scatters from the small particles. Since the large particles dominate the scattered intensity in this q regime, we conclude the average spacing between large particles does not change upon addition of the small particles. Increasing the number of the small particles in the system decreases the long range order displayed by the large particles. This is shown by the broadening of the six inner intensity maxima as the particle number ratio (Nbig/Nsmall) is decreased. To quantify the degree to which the spots are degraded, a procedure that approximates an angular autocorrelation function was developed. The 2D scattered neutron intensity patterns are angularly averaged over the entire q range, and the reciprocal lattice vector with the highest intensity is found. An annulus of 5 pixels on either side of the maximum intesity q vector is cut and divided into 72 5° bins. The neutron strikes in the bins are counted, and the intensities are nomalized to that of the bin with the most strikes. The function, referred to as the order parameter, OP60, is defined as
72
OP60 )
(
I(θi) I(θi+12) ∑ i)1
)
∑I(θi)2 72
(2)
Where I(θi) is the normalized number of neutron counts in the ith bin. This autocorrelation function provides a useful measure of the amount of sixfold symmetry in the scattering pattern. This definition is similar to that previously used but provides a higher angular resolution.21 There are two limiting cases for the patterns, uniform rings of an amorphous or polycrystalline sample and δ function intensity maxima indicating a high degree of long range, orientational order of randomly stacked hexagonal planes. For a uniform ring, the intensity in each bin is the same and the order parameter yields a value of zero. The case of the δ function intensity maxima has all bins with zero intensity except for 12 bins with intensities of 1. These are located in pairs, one on either side of the rays at 30°, 90°, 150°, 210°, and 270°. This is the approximation of a δ function which would be the result
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Figure 2. Scattered neutron intensities in the kv-ke plane for various number density ratios of big (B) and small (S) particles at equilibrium: (a) pure big particles; (b) 8.5 B/S; (c) 4.1 B/S; (d) 2.5 B/S. Colors moving from blue to red to yellow to white indicate more neutron strikes. All patterns are normalized to 107 incident neutrons. The center of each image is covered by a beam stop.
if the resolution was continuous. The order parameter from this configuration has a value of 10. All others cases will lie between these two bounds, with larger values of the order parameter indicating a higher degree of order. There are two central points to note in the SANS patterns for the mixtures under shear. First, when the pure large particles exhibit a ring of uniform intensity, all of the mixtures also exhibit a similar pattern. This can be seen in the order parameter plots (Figure 3) for the shear rates of 1 and 100 s-1 (polycrystalline and shear melted states of the pure large particles, respectively); the value of the order parameter stays near 0.3 for all mixtures as well as the pure large particles. From this we conclude that, for the range of shear rates investigated, the addition of small particles does not enhance the long range order of the pure large particles under nonequilibrium conditions. Second, at the shear rate where the pure large particle suspensions display long range order (at rest or in the sliding/shearing layer structures), a more dramatic effect is seen in the scattering patterns of the mixtures. At equilibrium, the pure large particles display densely packed planes of particles parallel to the walls of the gap
and the characteristic six intensity maxima pattern. The addition of a low density of the small particles has little effect on the scattered neutron intensities. The order parameter stays at roughly a constant value until a number ratio of about five big particles to every small particle is reached. As the densitity of small particles is raised above this value, the order parameter rapidly drops to the level of the amorphous samples, indicating a loss of the long range order in the system. A similar effect is seen with the systems under shear at 10 s-1. 3.2. Rheology of Mixtures. Accompanying the changes in microstructure are changes in mixture rheology. The effects of mixing are not as sudden, but the changes in flow properties are dramatically different for the structured suspension of the large particles and amorphous mixtures. Samples were subjected to the same shear history as used in the SANS experiments. At large applied stress, all of the mixtures as well as the pure large particle suspension approach a high shear rate limiting viscosity of roughly 0.02 Pa‚s. Note that at this stress level all samples display amorphous SANS patterns but no shear thickening is observed.
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Figure 5. Flow curves comparing the viscosity of the unmixed, salted, and highest mixing ratio latices. Note the viscosity of the salted latex still displays yielding characteristics not the zero shear rate plateau viscosity of the mixture.
Figure 3. OP60 vs number density ratio for all shear rates. The number of small particles in the system increases moving from right to left on the abscissa. The infinite number ratio is a suspension of big particles with no added small particles.
a KCl solution was prepared that when added in the same proportion as the small particles would contribute the same amount to the solution conductivity at a big to small number ratio of 2.0:1, as seen in Figure 5. The behavior is similar to that of the stock big particles with the dynamic yield stress still evident but at a lower value due to the decreased volume fraction.23 There is no evidence of a zero shear rate viscosity displayed by the salted stock suspension. We conclude that while the system is very close to the order/disorder transition and the added ions will bring the system closer to this boundary, the observed changes in viscometric properties are due to changes is the microstructure of the suspension of the large particles brought on by interactions with the small particles. 4. Conclusions
Figure 4. Flow curves for the various mixtures investigated.
For the stock solution of large particles, the viscosity diverges as the stress approaches zero (see Figure 4). Plastic deformations still occur, but steady state deformations are not observed. This material displays a dynamic yield stress, τdy, of about 0.01 Pa. As the number of small particles is increased, the τdy disappears and a zero shear rate plateau viscosity is observed. The exact location of the loss of the yield stress is not clear. However, the zero shear rate viscosity can be clearly defined at a large/small particle density ratio of 2:1, which is close to that where the ring patterns are fully developed in the equilibrium structure (compare with Figure 3). From this we conclude that the unmixed large particles behave as a weak viscoelastic solid. Upon the addition of a sufficient quantity of small particles the suspensions behave like a fluid. It is possible that the addition of the small particles to the mixture raises the volume fraction of the order/disorder transition by raising the ionic strength (i.e., simply adding the ions that exist in the small particle suspension is sufficient to produce melting). To examine this possibility,
At constant total volume fraction, as small particles are added to an ordered suspension of large particles just above the order/disorder phase transition boundary, long range order is lost. As the order is destroyed, despite there being no changes in the average large particle spacing, the suspension loses its yield stress and becomes a shear thinning fluid with limiting high and low stress viscosities. Simple dilution of the suspension with deionized water or with an electrolyte solution which contributes the same effective ion concentration as accompanies the small particles is insufficient to produce the observed behavior. Several conclusions can be drawn from this study. First, just above the order/disorder phase boundary long range order is required to produce a viscoelastic, solid-like response. Whether this conclusion would hold far from the order/disorder phase boundary of the large particles is unclear. However, for these suspensions, an ordered structure results in a viscoelastic solid while a disordered suspension at the same volume fraction is a fluid. Second, the crystal of large particles can sustain a range of smallparticle “defects” prior to melting. Complete melting occurs over a narrow range of small-particle number density. Third, in suspensions that have a steady state polycrystalline microstructure at low shear rates, long range orientational order is re-established at intermediate shear rates and all suspensions shear melt at higher shear rates. This last transition occurs with no indication of shear thickening. (23) Hunt, W. J. Studies on the Rheological and Microstructural Behavior of Charge-Stabilized Colloidal Particles. M.S. Thesis, University of Illinois, 1996.
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Acknowledgment. We thank Dr. H. Hanley, Dr. G. Straty, Dr. C. Glinka, and Dr. J. Barker of NIST for their assistance in running the SANS experiments and for helpful discussions. We acknowledge the support of the National Institute of Standards and Technology, Department of Commerce, in providing the facilities used in this experiment. This material is based upon activities supported by the National Science Foundation under Agreement No. DMR-9122444. The electron microscope
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work was carried out in the Center for the Microanalysis of Materials, University of Illinois, which is supported by the U.S. Department of Energy under Contract Number DE-AC 02-76ER 01998. This work was supported by the Department of Energy through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign. LA960416J
