Langmuir 1994,10, 2817-2829
2817
Rheological and Microstructural Transitions in Colloidal Crystals L. B. Chen,' M. K. Chow,$ B. J. Ackerson,s and C. F. Zukoski*?t Department of Chemical Engineering and Beckman Institute, University of Illinois, Urbana, Illinois 61801, and Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078 Received October 25, 1993. I n Final Form: April 20, 1994@ The flow of suspensions of 229-nm, charge-stabilized,polystyrene latex particles is directly linked with suspension microstructure. Transitions from long-range orientational order at rest to a steady-state, polycrystalline phase are associated with exceeding the dynamic yield stress. Application of stresses larger than a critical value nucleates a sliding layer microstructurewhich is accompanied by a discontinuous drop in suspension viscosity. Over a range of shear rates, polycrystalline and slidinglayer microstructures coexist. At elevated shear rates, stresses and microstructures fluctuate in time. Shear thickening accompanies the complete loss of long-range orientational order. The transitions in rheology and microstructure are reversible except at very high shear rates where shear thickening accompaniesirreversible flocculation.
I. Introduction Links between rheology and flow-induced microstructures in colloidal suspensions have been investigated for a number of systems including amorphous suspensions of hard spheres1 and uniform2-12 and bimodal suspensionsl1J2 of charge-stabilized particles. While nonequilibrium simulations of particles experiencing soft interactions show structures similar to those seen experimentally, linking these simulations to the experimental systems remains p r ~ b l e m a t i c . l ~Few - ~ ~ experimental studies have simultaneously determined microstructures and rheological behavior, further increasing the difficulty in making corrections between experiments and models. In this paper, we describe the flow properties and microstructures of two suspensions of 229-nm-diameter, charge-stabilized particles at volume fractions where the suspensions are ordered at rest. We report optical and small-angle neutron scattering microstructural evidence +
University of Illinois.
* Oklahoma State University. @
Abstract published in Advance ACS Abstracts, June 15,1994.
(1)Ackerson, B. J. J . Rheol. (N.Y.) 1990,34 (41, 553.
(2) Krieger, I. M. Adu. Colloid Interface Sci. 1972,3,111. (3) Buscall, R.; Goodwin, J.; Hawkins, M.; Ottewill, R. H. J . Chem. SOC.,Faraday Trans. 1 1982,78,2889. (4) Tomita, M.; Van de Ven, T. G. M. J . Colloid Interface Sci. 1984, 99 (21, 374. ( 5 ) Ackerson, B. J.; Hayter, J. B.; Clark, N. A.; Cotter, L. J . Chem. Phys. 1986,84 (41, 2344. (6) Ackerson, B. J.; Clark, N. A. Phys. Rev. A 1984,30 (21, 906. (7) Goodwin, J . W.; Gregory, T.; Miles, J. A,; Warren, B. C. H. J . Colloid Interface Sei. 1984,97,488. (8) Chen, L. B.; Zukoski, C. F. J . Chem. Soc.,Faraduy Trans. 1990a, 86 (141,2629. (9) Chen, L. B.; Zukoski, C. F. Phys. Reu. Lett. 1990b,65 (l), 44. (10)Chen,L. B.;Zukoski, C.F.;Ackerson,B. J.;Hanley,H.J.;Straty, G. C.; Barker, J.; Glinka, C. J. Phys. Rev. Lett. 1992,688. (11)Hanley, H. J.;Pieper, J.; Straty, G. C.; Hjelm, R. P., Jr.; Seeger, P.A. Faraday Discuss. Chem. SOC.1990,90,91. (12) Lindsay, H. M.; Chaikin, P. M. J . Chem. Phys. 1992,76 (71, 3774. (13) Ashdown, S.; Markovic, I.; Ottewill, R. H.; Linder, P.; Oberthur, R. C.; Rennie, A. R. Langmuir 1990,6, 303. (14) Hess, S. Int. J . Thermophys. 1986,6, 657. (15) Loose, W.; Hess, S. Rheol. Acta 1989,28 (21, 91. (16) Woodcock, L. V. Chem. Phys. Lett. 1984,111, 455. (17)Robbins, M. 0.;Kremer, K.; Grest, G. S. J . Chem. Phys. 1988, 88 (51, 3286. (18) Bonnecaze, R. T.; Brady, J . F. J . Rheol. (N.Y.) 1992,36(11, 73. (19) Barnes, H. A,; Edwards, M. F.; Woodcock, L. V. Chem. Eng. Sci. 1987,42,591. (20) Hanley, H. J. M.; Morriss, G. P.; Welberry, T. R.; Evans, D. J. Physica A 1988,149,406.
0743-7463/94/2410-2817$04.50/0
showing a direct link between particle packing and flow properties. As in a previous report,1° we find that while displaying long-range orientational order at rest, these suspensions have steady-state polycrystalline structures at low shear rates. As the shear rate is increased, longrange orientational order is reestablished over a wide range of shear rates prior to shear melting where all longrange order is lost. The transition from a highly ordered state a t rest to polycrystals is associated with the application of a stress in excess of the dynamic yield stress while the reestablishment of long-range orientational order is associated with a critical stress and metastable viscosities. Finally as shown in many previous studies, the loss of long-range orientational order accompanies shear t h i ~ k e n i n g . ~ l - ~ ~ In this study, steady-state polycrystalline phases have been explored and are found to represent the steady-state microstructure over a wide range of shear rates. The transition from polycrystals to long-range orientational order at elevated shear rates is reversible on increasing or decreasing shear rate. The transition from polycrystals to long-range orientational order a t rest is very slow. However, subjecting the polycrystalline sample to oscillatory strains induces long-range orientational order as is observed in suspensions of hard spheres. While microstructures and flow properties have been studied separatelyin a number of studies, there have been few reports where both have been simultaneously determined.10,21s22,24 Here we continue to bridge this gap and shed light on the nature of nonequilibrium microstructural phase transitions. The experimental system is described in section 11, prior to detailing both rheological and microstructural data in section 111. Attention is given to stresses and shear rates where microstructural and rheological transitions occur such that comparisons with (21) Hoffman, R. L. J . Colloid Interface Sci. 1974,46 (3), 491. (22) Laun, H. M. Prog. Trends Rheol. II 1988,287. (23) Laun, H. M.; Bung, R.; Schmidt, F.; J . ofRheology 1991,35(61, 999
(24) Laun, H. M.; Bung, R.; Hess, S.; Loose, W.; Hess, 0.;Hahn, K.; Hldicke, E.; Hingmann, R.; Schmidt, F.; Lindner, P. J . Rheol. 1992,36 (41, 743. (25) Boersma, W. H.; Laven, J.; Stein, H. N. MChE J . 1990,36(31, 321. (26) Boersma, W. H.; Baets, P. J . M.; Laven, J.; Stein, H. N. J.Rheol. 1991,35 (61, 1093. (27) Barnes, H. A. J . Rheol. 1989,33,329.
0 1994 American Chemical Society
2818 Langmuir, Vol. 10, No. 8, 1994
Chen et al.
other systems can be made. The results are discussed in section IV and conclusions drawn in section V.
Stator
-
Radius R
-
59.98 mm
11. Experimental Section Suspensions investigated were composed of particles prepared by the method ofHomola and James.28 These particles were 229 h 2.4 nm in diameter as determined by sizing over 100 particles from transmission electron micrographs. Particle diameters were confirmed with dynamic light scattering, suggesting the polymer hairs from the ionic comonomers used in synthesis do not substantially alter the hydrodynamic diameter from the transmission electron microscopy (TEM) diameter. Particles were cleaned following the sedimentatioddecantation procedure suggested by Harding and Healy.29 Centrifugation at l l O O O g for 2 h during each cycle was sufficient to completely pack the particles to the bottom of the centrifuge tubes. For the first five cycles, M HCl was used t o redisperse the latex cake. Subsequent resuspension was carried out with solutions of M KCl. High volume fractions were obtained by placing the dilute latex in freshly boiled dialysis membranes and applying 2 psig of nitrogen when the sacks were submerged in M KC1. After 6 - 10 days, suspension volume fractions reached values ranging from 0.3 to 0.6. Suspensions were stored in dialysis sacks submerged in M KC1. The dialysate was checked at regular intervals over a 3-month period until there were no changes. After 3 months, the suspensions were placed in tightly sealed plastic bottles. Rheological measurements were preformed with Bohlin constant stress (CS) and variable rate of strain (VOR) rheometers. A cup and bob geometry was used where the bob had a diameter of 25 mm and a gap spacing of 1.2 mm. Steady-state shear rate and creep and recovery measurements were made using the CS. Steady-state shear rate results were compared to VOR results. At a given shear rate, stresses were in reasonable agreement W%).Errors in absolute stress readings with the VOR occur due to difficulties in establishing an absolute zero stress for these suspensions with large yield stresses. In order to observe structural changes within the suspension, an outer cup made from 3-mm-thick fused silica was used in the CS. After placing the sample into the cup, the bob was lowered, and red gauge oil (Fisher Scientific) was then pipetted onto the top to minimize evaporation. Extensive studies have shown that this procedure does not alter suspension flow behavior over periods of time longer than 4 weeks. Samples were initially subjected to modest shear (150 s-l) for 5- 10 min and then allowed to rest for time periods exceeding 3 h. Creep and recovery curves were measured with applied stresses of 0.01-50 Pa. The control software supplied with the CS allows it to be operated in a constant shear rate mode. Due to long relaxation times of the samples investigated, 3600 s was allowed for the sample to reach steady state prior to determining the stress and moving to the next shear rate. The lowest possible shear rate that could be applied with this technique was When the fused silica cup was used, structural changes could be observed optically. A Minolta 35-mm camera with a 50-mm lens was used to photograph structural changes as a function of flow history. Small-angle neutron scattering (SANS)was carried out at the Cold Neutron Scattering Facility at the National Institute of Standards and Technology in Gaithersburg, MD. Beam lines 8 and 30 m in length were used. For the 8-m line a pinhole collimation system illuminated the sample and a 1.5-nmneutron wavelength was used. The array detector was fixed 3.6 m from the sample, giving a wave vector range of 0.03-0.57 nm-l. Investigations with the 30-m SANS facility involved neutrons with a wavelength of 1.5 nm and a detector placed 15 m from the sample, giving a wave vector range of 0.015-0.14 nm-'. Both beam lines were equipped with two-dimensional positionsensitive detectors with a square matrix of 128x128 detector elements each with an area of 26.01 mm2. The time for each scan was normally 10 min while some scans were made for more than 30 min. The reported intensities are thus time-averaged measurements. ~~
(28) Homola, A,; James, R. 0. J. Colloid Interface Sci. 1977,59(11, 123. (29) Harding, I. H.; Healy, T. W. J. Colloid Interface Sci. 1982,89 (11,185.
1
p ko
X-Rsine,
Figure 1. Geometry for scattering experiments. In (a, top), a collimated neutron beam impinges on the Couette shear cell and the scattered neutrons are collected on a detector 3.6 or 15 m from the cell. In (b, bottom left) and (c, bottom right), a top view of the shear cell is shown with the beam passing through the shear cell in two different orientations. The scattered wave vector, k, is the difference between the incident, ko, and scattered, k,,wave vectors. The SANS shear cell is made from fused quartz consisting of a Couette geometry with the outer cylinder rotating. The shear cell has an inner diameter of 59.98 mm and an annular gap of 0.49 mm. The neutron beam is scattered from the suspension twice, once on entering and once on leaving. The shear cell can be driven at shear rates of 4 x 10-3-3 x lo4s-l with a precision of 0.1%. The scattering geometry is similar to that reported earlier,'O with the detector giving scattering vectors k = k& + k$,. The magnitude of k is given by (4n/i) sin(W2)where i is the neutron wavelength (1.5 nm) and 0 is the scattering angle. The vertical ory direction with unit vector by is always pointed in the direction of the vorticity vector in the shear cell, giving k, = k, and = G,. The horizontal or 2 direction scattering vector (unit vector of &,I can consist of two components depending on where the beam passes through the shear cell. If the beam passes through the center of rotation, k, = k, and GX = 0, where k,&, is the scattering vector directed along the direction of the velocity in the shear cell. Here b, is the unit vector in the velocity direction. If the beam is moved off axis as shown in Figure 1, k, takes on a component from the scattering vector in the direction of the shear gradient, Go, such that k&, = kudu+ k d o . As shown in Figure 1,k, = k, sin Bc where 0, is the off-axis angle. Intensities were analyzed by averaging neutron counts over several detectors in the vicinity of an intensity maximum. This approach reduces noise in the data but also lowers the resolution of results. As a consequence, we report structures consistent with the observed scattering patterns but are unable to make unambiguous structural assignments. As discussed below, structural determinations were made by exploring the modulation of intensity maxima along .2, For angles Be > 64", the signal was lost due to reflection of the neutron beam by the quartz cell. Neither the optical nor the SANS studies are capable of providing unambiguous information about the velocity profile in the rheometer tool gap. Rheological and structural studies on suspensions of volume fractions of 0.45 and 0.53 are reported here. Extensive rheological characterization at other volume fractions of these particles
Langmuir, Vol. 10, No. 8,1994 2819
Rheological and Microstructural Transitions
e
I
i m
e
--
Figure 2. Optical photographs of the shear cell with different shear histories: (a, top left) # = 0.53,at rest, (b, top middle) 4~= 0.45,1 h after a stress of 1.2 Pa has been applied, ( c , top right) # = 0.53,p = 0.019 s - ~ ,(d, bottom left) # = 0.53, P = 2-43S-l, (e, bottom middle) # = 0.53,P = 6.05 s-I, (f, bottom right) 4 = 0.53,p = 19 s-l. indicate that the order/disorder volume fraction for the suspensions is near 0.37. The suspensions under consideration had osmotic pressures of 138 and 303 Pa for volume fractions of 0.45 and 0.53,respectively, as determined by measuring the column height required to bring suspensions of these volume fractions M KCl across a dialysis membrane.30 into equilibrium with
111. Results After shearing for 10 min at 150 s-l and resting the samples for over 1h, they had an opalescent appearance (30) Chen, L. B. Ph.D. Thesis, University of Illinois, 1991.
(Figure 2a) and showed long-range orientational order in SANS experiments (Figure 3a). For stresses less than the static yield stress, zys,the appearance of the suspension was not altered, and on release of the stress, all strain produced upon application of the stress was recovered (Figure 4). For z > rys,the suspension did not reach a steady-state rate of strain in time periods of 5-10 ks and not all strain observed on application of the stress was recovered on release of the stress (Figure 4). For z < zys, an elastic modulus, Go, could be determined from a recoverable straidstress curve. The recoverable strain,
2820 Langmuir, Vol. 10, No. 8, 1994
Chen et al.
Figure 3. SANS scattering patterns with 8, = 0 at various shear rates: (a, top left) at rest, (b, top middle) 0.08 s-l, (c, top right) 0.3 s-l, (d, middle left) 15 scl, (e, middle middle) 40 s-l, (f, middle right) 500 s-l, (g, bottom) 6000 s-l.
yr, is the snap-back strain achieved after the applied stress is set to ~ e r o . ~ , Oscillatory ~O strain measurements were also used to determine the suspension's elastic modulus. For strains in the linear viscoelastic region, the storage and loss moduli showed no frequency dependence in a range of 10-3-10 Hz. Both creep and recovery and oscillatory strain techniques for determining Go agreed, giving 55.5 and 214 Pa for Go for the suspensions with volume fractions of 0.45 and 0.53, respectively. Limiting linear viscoelastic strains (i.e., the strain above which, in an oscillatory strain experiment, the modulus is no longer independent of strain) and the static yield stresses for the suspensions are given in Table 1. For stresses above zys,the suspension's appearance was altered during creep and recovery experiments. The smooth iridescence seen at rest was broken into mottled zones as if the crystal was forming shear zones. The furrows or lines of different color developed in chevron patterns with a characteristic angle of approximately 7080". Figure 2b shows a suspension exposed to 1.2 Pa for 1 h where the striation pattern occupies approximately half the surface area of the shear cell. On release of the stress, the striation patterns persist for times longer than
3 h. These patterns were only seen for zys < z < zydand, as a result, could not be observed with SANS where only continuous rates of deformation could be applied. As the stress is further raised above zys,the recoverable strain becomes a nonlinear function of applied stress (Figure 5 ) and the striation pattern expands to occupy the entire observable surface. For example, after stressing the same suspension pictured in Figure 2c a t 1.6 Pa for 10 ks, the striations cover the entire surface but the chevrons are still observable. Note the striations are still iridescent. At a characteristic stress denoted the dynamic yield stress, zyd,a steady-state rate of strain is reached 500 s or less after applying the stress. This contrasts markedly with the sluggish behavior seen for z < zyd. At zyd,the recoverable strain curves for both suspensions change in slope. The recoverable strain at zydfor both suspensions was near 0.04 (Figure 5 , Table 1). As the stress is raised above zyd,the stress-steady state shear rate curves show a low shear rate plateau stress (Figures 6 and 7). The reported shear rates are apparent and do not necessarily reflect that the suspension is deforming with a uniform shear rate. The open circles in Figures 6 and 7 denote
Langmuir, Vol. 10,No. 8, 1994 2821
Rheological and Microstructural Transitions
.&
zc -i l ll
I
$
0.5 Pa
0
2
4
6
I
8
10
12
time (ks) Figure 4. Creep and recovery compliance ( y ( t ) / t ) as a function of time for two stresses applied to the suspension at 4 = 0.53. For t = 0.5 Pa, the sample shows elastic deformation and complete recovery of a solid. At t = 3 Pa, the suspension shows the slow creep response and nonrecoverable strain characterizing behavior for tr* < t < tyd. steady-state stress when the system is driven at a given shear rate, y , and the shear rate increases after each measurement. The open squares denote the steady-state stress measured when the system is driven at a given shear rate in a decreasing shear rate ramp while the closed circles indicate steady-state shear rates when the sample is driven at a given stress. In the low shear rate region, the suspensions take on a distinct polycrystalline appearance (Figure 2c). The surface of the shear cell is covered with small colorful domains which do not change appearance substantially with further shearing. SANS patterns show a loss of the original long-range order as noted by the absence of secondary scattering maxima and the substantial broadening of spots in the primary scattering ring. Polycrystalline microstructures with some retention of crystallite orientation are indicated by the nonuniformity of the intensity in the primary scattering ring (Figure 3b). Similar scattering patterns were observed with the sample of volume fraction 0.45 on the 8-m SANS line. However, the intensity maxima were not discernable in these experiments. With increasing k,, the SANS pattern becomes a spotty ring, suggesting a common structure in strained crystals (Figure 8)and the long-range order seen at rest. With increasing shear rate, the stress slowly increases to a maximum value and then decreases. The maximum value is denoted t* and is reached at a shear rate of pL*. In constant shear rate experiments, z* denotes the beginning of the anomalous flow region where the stress is a decreasing function of shear rate. The stress drops. to a minimum value at p, before starting to monotonically increase again. The stress decreases 1 Pa for 4 = 0.53 and 0.3 Pa for 4 = 0.45. y , is reached at 0.319 and 0.202 s-l for 4 = 0.45 and 0.53, respectively. In the stress minimum, suspensions retain a polycrystalline appearance when observed optically. In SANS studies, the scattering pattern remains polycrystalline in nature with the emergence of a weak four-spot pattern. Off-axis studies show this pattern evolving to one with six primary intensity maxima rotated 30" from that seen at rest (Figure 9). Note that the spots are broad and the secondary scattering ring retains a uniform intensity. Due to the lower resolution of the data taken with the 8-m SANS, structural details at this level could not be seen.
However, a polycrystalline pattern was retained at shear rates near the stress minimum.1° In constant-stress experiments, t* denotes a point where the steady-state suspension viscosity discontinuously decreased an order ofmagnitude (Figures 10 and 11).Note that the viscosities reported are defined as the ratio of the stress and the apparent shear rate. The lower and higher shear rates seen at t* are y~ and y ~respectively. , If the suspension is sheared at t < z* for 30 min and then the stress is stepped to values just above z*, the bob reaches a pseudo steady state before acccelerating to the true steady state. Metastable shear rates and viscosities (shown as stars in Figures 6 and 7) are observed by following the creep compliance as a function of time. A n example of this response is shown in Figure 12 for 4 = 0.53 where the creep compliance ( y ( t ) / t )is given in Figure 12 for the suspension subjected to 10.85 Pa (for 4 = 0.53 where t* = 9.3 Pa). After a short elastic response time, a pseudo-steady-state rate of strain is reached at 1.0 s-l corresponding to a viscosity of 10.8 Pa s. After 10 s, the bob accelerates to the true steady-state shear rate of 9.9 s-l corresponding to a viscosity of 1.09 Pa s. Similar behavior is seen for 4 = 0.45. For t < t*, the suspension has a polycrystalline microstructure. As shown in Figure 13,the discontinuous change in viscosity seen for t > t* is associated with a dramatic change in the optical appearance of the suspension. As the bob accelerates from the metastable shear rate, the polycrystalline domains are replaced by large zones of smooth iridescence. For z just larger than t*,a small fraction of the surface is covered with smooth zones at steady state. As t grows, the surface area covered by polycrystals shrinks and the regions of uniform color grow. During this process, the time required to nucleate the lower viscosity phase in creep experiments decreases from over 1 h near t* to less than 1 s when the surface is completely covered by a smooth, uniform color.8 In SANS experiments, as the shear rate is increased out of the stress minimum the scattering pattern begins to show a greater degree of order. Note in Figure 3d that the secondary scattering ring is breaking into distinct scattering maxima while the primary scattering maxima become sharper. In the shear rate region where a mixture of polycrystalline and smooth zones is seen optically, the SANS pattern continues to show broad peaks which become sharper and more distinct as the shear rate is raised. Evidently, the discontinuous drop in viscosity seen in constant-stress experiments is associated with the nucleation and growth of this high shear rate ordered phase. Polycrystalline and long-range orientationally ordered phases coexist over a range of shear rates spanned approximately by the stresses where metastable viscosities can be observed. As the shear rate is increased above y , the suspension viscosity decreases up to a shear rate ycwhere the viscosity becomes an increasing function of shear rate. At 4 = 0.53, kc lies above 700 s-l and could not be probed using the standard rheometer tools due to the onset of Taylor in~tabilities.~ Instead, ~ a high shear cup with a very narrow gap was used to probe the rheology at high y . The high shear cup used consists of two concentric, tapered cones. The gap between the operating surfaces is set by the vertical displacement between the two cones. A gap of 7 pm was used in the data presented here. As reported elsewhere,32the location of yc depends weakly on gap spacing, with pc decreasing for smaller gaps. These investigations suggest that, for y c: ye, suspension vis(31) Bird, R.;Stewart, W. E.;Lightfoot, E.N. Transport Phenomena; John Wiley and Sons, Inc.: New York, 1960. (32)Chow, M. K. Ph.D. Thesis, University of Illinois at UrbanaChampaign, 1993.
2822 Langmuir, Vol. 10, No. 8, 1994
Chen et al.
Table 1. Summary of Rheological Propertiesa
4
J C , Pa ~
Go,"Pa
tys,dPa
yyse
tyd,fPa
yrdg
t*,h Pa
y ~ s-l :
~ H s-l S
0.45 0.53
138 303
55.6 214
0.61 2.0
0.011 0.009
2.0 7.5
0.045 0.043
2.44 8.95
0.10 0.11
1.04 1.30
y*
Pa
yc,l s-l
tc,m
600
30
0.048 0.046
a 229-nm latex particles dialyzed to equilibrium against M KC1. Osmotic pressure. Zero frequency modulus of the presheared crystal with strain applied to hexagonally close packed planes along the direction of closest packing. Static yield stress (for t < rys,no f Dynamic yield stress (for t < tyd, steady-state plastic deformation occurred on application of stress for lo4 s). e Recoverable strain at tys. rates of strain were not achieved lo4 s after applyingstress; for t t tud,steady-state rates of strain were reached 500 s or less after applying strain).g Recoverable strain at -cyd. Stressmarkingdiscontinuousdrop in viscosity. Low shear rate at t*.1 High shear rate at t*. Recoverable strain at t*. Shear rate marking onset of temporal fluctuations in stress. Note that abrupt shear thickening was not observed until the shear rate was stepped from 500 to 6000 s-l. Stress at -yc.
0.06
-
0
c .&
0
(d
L
8
3 v)
0.04 -
Q,
s
0
(d
k
Q,
-
3
$ 0.02 -]
P, L
0
7
O /
I ' 0.00
0
2
6
4
8
10
shear stress (Pa) Figure 5. Recoverablestrain, yr, as a function of applied stress at volume fractions of 0.45( 0 )and 0.53 (0).The lines drawn at low stresses indicate the limit of the linear region and are used to determine GO. 45 3
4 0
-
o 0
-a
3.5
-
Bb
$
a
1.5
5 io
-D
io
i
shear rate
10
lo'
1 10'
(s-l)
Figure 7. Shear stress as a function of shear rate at a volume fraction of 0.53. Symbols have the same meaning as in Figure 6.
period of time, the stress would undergo a step change to a new steady-state value. This is illustrated in Figures 15 and 16 where step changes in time-averaged stresses are apparent between 1000 and 1500s after a step change in shear rate from 500 SKI. If the maximum shear rate was held less than 5800 s-l, the suspension could be cycled between time-invariant stresses at 500 s-l and fluctuating stresses a t higher shear rates, demonstrating that the fluctuations involved no irreversible damage to the suspension. If, however, the shear rate was stepped from 500 to 5800 s-l, the sample irreversibly flocculated. Shown in Figure 14 are the minimum and maximum stresses measured in a 2500-s time period (sampling every 5 min) as well as the time-averaged stresses for the high and low plateau stresses. Each data point was measured after equilibrating the suspension a t 500 for 5 min and then stepping up in shear rate. Note the minimum and time-averaged low plateau stresses superimpose at 3600 and 4800 s-l. As the shear rate increases, the high and low plateau values, as well as the maximum and minimum stresses, diverge. The fluctuating stresses suggest variations in microstructure, and this was confirmed in SANS studies where scattering patterns averaged over 5 min varied between patterns with well-defined primary and secondary scattering maxima and patterns displaying little long-range orientational order. Time variations in scatteringpatterns are displayed in Figure 17 where the sample had been equilibrated a t 500 s-l for 5 min (yieldinga SANS pattern as shown in Figure 3 0 prior to stepping to 3000 s-l. As shown, over a time period covering 30 min, long-range orientational order is apparently lost and regained. Microstructural fluctuations diminished as the shear rate was raised until only an amorphous scattering pattern was seen at 6000 s-l (Figure 3g). In the Couette shear
i lo-'
Y
10
10
(I/&
Figure 6. Shear stress as a function of shear rate at a volume fraction of 0.45 determined from constant shear rate measure-
ments with increasing shear rate ( 0 )and decreasing shear rate determined at constant applied stress are indicated by 0 , while a metastable shear rate (see text) is indicated by Ir. (0). Steady-state shear rates
cositites are independent of gap down to tool spacings of 25 particle diameters. Data taken with the tapered plug rheometer tools are shown in Figure 14. Relatively good agreement is found for the region where concentric cylinder and tapered plug shear rates overlap. Above a shear rate of 500 s-l, the stresses measured with the tapered plug began to show temporal fluctuations. With increasing shear rates, the magnitude of the fluctuations increased. Often, after a
Rheological and Microstructural Transitions
Langmuir, Vol. 10,No. 8, 1994 2823
Figure 8. SANS patterns of the suspension at a volume fraction of 0.53 at p = 0.08 s-l and off-axis angles of (a) 8, = 0" (on axis) and (b) 8, = 24".
Figure 9. SANS scattering patterns for the suspension with $J = 0.53 at p = 0.3 s-l at off-axis angles of (a) 8, = 0" (on axis) and (b) 8, = 60".
cell used in SANS studies, long-range orientational order was recovered on reducing the shear rate to 500 s-l for shear rates of 6000 s-l and below. At higher shear rates, irreversible flocculationoccurred. The stress fluctuations, shear thickening,and loss of long-range orientational order observed here are in keeping with previous studies on suspensions of uniform spheres.21r23>26 The range of shear rates where instability in the flow curves and disordering seen in SANS studies occur covers an order of magnitude variation in p. As with the polycrystalline and sliding layer microstructures, the sliding layer and amorphous structures appear to coexist over a broad shear rate range, suggesting a gradual rather than abrupt microstructural transition. In summary we find that if presheared, at rest both volume fractions have long-range orientational order which gives a characteristic SANS pattern of a primary hexagon with a peak oriented in the direction of the vorticity vector in the shear cell. When a stress larger than the static yield stress is exceeded, the long-range orientational order is gradually replaced by a polycrystalline microstructure. Above the dynamic yield stress, steady-state rates of strain are measured and suspensions are observed visually t o have a polycrystalline microstructure with weak residual crystallite orientation seen
10 -z 1.5
2.0
2.5
3.0
3.5
4.0
4 5
s h e a r s t r e s s (Pa) Figure 10. Viscosity as a function of shear rate at 4 = 0.45.
Symbols are the same as those in Figure 6. in SANS. In constant shear rate experiments, a local stress maximum occurs a t z* and the stress becomes a
Chen et al.
2824 Langmuir, Vol. 10, No. 8, 1994
lo
polycrystals seen optically (Figure 18). The strain and frequency dependence of this phenomenon has not been explored in detail, but these observations are reminiscent of shear-induced ordering reported in hard sphere suspensions.'
' L lV. Discussion
00 O O 0
0
0
0
10 5
10
15
20
25
shear stress (Pa) Figure 11. Viscosity as a function of shear rate at 4 = 0.53. Symbols are the same as those in Figure 6. 10
e
10 '
Recently we reported the appearance of polycrystals and the links between microstructural and rheological transitions seen a t t* for the suspension at a volume fraction of 0.45.1° More detailed scattering and rheological studies are reported here for 4 = 0.45 and 0.53 composed of suspensions of the same particles. First we briefly discuss the structural assignments made for the suspensions showing long-range orientational order and make comparisons between these samples and previous reports of shear-induced order in charge-stabilized suspensions. Following Pusey et al.33and Loose and A ~ k e r s o nthe ,~~ structures of the suspensions show long-range orientational order and can be modeled as consisting of sheets of hexagonally close packed particles. If the suspensions are illuminated normal to the vorticity/velocityplane with lines in the close-packed direction (Figure 19)aligned with the velocity vector, the scattering pattern will consist of a hexagonal array of scattering maxima oriented as in Figure 3a. In scattering vector space, the intensity maxima will lie along lines of constant k , and k,. When these planes are then stacked along the k , direction (out of the page in Figure 19),the basic form of the scattering pattern is not altered. Intensity maxima will occur at the same values of k , and k , as for a single plane. However, the intensity maxima will be modulated along k,. Thus, when the neutron beam is passing through the shear cell's axis of rotation, the points on the scattering plane can be described as k = k,&, k,&, 06, and the SANS pattern reflects symmetry in the velocity/vorticity plane. When the beam is pulled off axis, nonzero k , components are introduced. By following the relative intensity of intensity maxima along k , for lines ofconstant k , and k,, information about the stacking sequence can be learned. We have analyzed our data for two intensity maxima, one in the primary ring, ql, and one in the secondary ring, 9 2 . ql has coordinates of [2nldl&, + [2n/(2/3d)l&, k,&, while q 2 = [2n/d]&, [2n(2/3)/d]&, kg,. Here d is the lattice parameter for the crystal given, d = u(4m/4)1cc where ois the particle diameter and &is the close-packedvolume fraction of 0.74. As discussed e l ~ e w h e r e ,if~the ~ - crystal ~~ were a perfect face-centered cubic crystal, ql would not be visible at k , = 0. 6 function maxima in ql(k,) would be seen at k , = (2n/d)0.41 and (2x/d)0.81, with all six intensity maxima in the primary ring having the same intensity. Maxima in qZ(k,) are expected at k , = 0 and (2n/d)1.225. For a perfect hexagonally close packed crystal these 6 function maxima in both the primary and secondary scattering maxima occur with k , = 0 and (2nl d)0.61. At shear rates above 40 SKI, the suspension shows longrange orientational order in the &-&, plane (Figure 30. This high shear rate, ordered scattering pattern has six primary scattering maxima, with the two along &, being of lower intensity than the four side peaks. As shown in Figure 20, ql(k,) and qz(kv)are weakly dependent on k , for y = 40, 500, and 1000 s-l. The roll-off in intensity with increasing k , can be attributed to decreases in the particle form factor (which is not divided out). The lack of modulation of q,(k,) and qz(k,) indicates the loss of
+
time (s) Figure 12. Creep compliance measured on the suspension at 4 = 0.53 for an applied stress of 10.85 Pa. The lines drawn at short and long times indicate where the log(comp1iance)-log(time) curve has a slope of unity. The arrows indicate times where photographs shown in Figure 13were taken: (a)6 s, (b) 60 s, (c) 110 s, (d) 300 s, and (e) 3600 s. decreasing function of shear rate until a shear rate of y , is reached where the stress once more monotonically increases with shear rate. In constant-stress experiments t* marks a discontinuous drop in viscosity and the nucleation of a high shear rate microstructure with longrange orientational order. Over a range of shear rates, zones of polycrystals and regions with long-range orientational order coexist. With increasing stress above t*, the viscosity continues to shear thin, and the intensity maxima seen with SANS sharpen. At still higher shear rates, the intensity maxima broaden, fluctuate in time, and are finally lost. The shear rate range where microstructures fluctuate corresponds quite closely with the shear rate region where temporal variations in stress are observed. The rheology and microstructures are reversible on increasing or decreasing the shear rate (for y 5 6000 s-l) except for the transition from polycrystals to longrange orientational order when the shear rate is set from a small value s-l) to zero. Polycrystals remain for time periods exceeding several days. However, if the suspension is subjected to oscillatory strains, long-range orientational order is restored as indicated by the loss of
+
+
+
+
(33)Pusey, P. N.; van Megen, W.; Ackerson, B. J.; Bartlett, P.; Underwood, S. M. Phys. Reu. Lett. 1989, 63, 2753. (34) Loose, W.; Ackerson, B. J. Manuscript in preparation. (35) Sirota, E.B.; Ou-Yang,H. D.; Sinha, S. K.; Chaiken, P. M. Phys. Reu. Lett. 1989, 62, 1524. (36)Chen, L. B.; Ackerson, B. J.; Zukoski, C. F. J.Rheol. 1994,38, 193.
Rheological and Microstructural Transitions
Langmuir, Vol. 10, No. 8, 1994 2825
Figure 13. Optical photographs of the shear cell a t different times after applying a stress of 10.85 Pa t o the suspension at 4 = 0.53 (the creep compliance curve is shown in Figure 12): (a, top left) 6 s, (b, top middle) 60 s, (c, top right) 110 s, (d, bottom left) 300 s, (e, bottom right) 3600 s.
correlationsin the C,, direction in particle positions between different layers.33734136 The increased intensity of ql(k,) over the primary scattering maxima along indicates that rows of particles oriented in the &, direction have centered between rows of particles in adjacent planes to ease slippage. This scattering pattern has been reported for numerous suspensions and is referred to here as resulting from the sliding layer structure. Note, however, that the uniformity of shear across the suspension cannot be determined from these studies. The SANS pattern reported on samples at rest (Figure 3a) was gathered by preshearing the suspension at 150 s-l after loading the sample into the shear cell prior t o setting p to zero. If preshearing was not carried out, the sample’s optical appearance and corresponding SANS pattern were those of polycrystals. Preshearing resulted in Figure 2a and a SANS pattern shown in Figure 3a. As discussed above, the preshearing produces hexagonally close packed layers lying parallel t o the rheometer wall with little registration of particles between adjacent layers.
When the shear rate is set to zero, these layers will relax in time t o the lowest energy configuration. Due to the elevated volume fractions studied, particle and layer movement toward greater layer registration will be extremely hindered. As a result, only weak layer registration is expected. As shown in Figure 20, ql(k,) and qz(k,) show weak modulation with increasing k,. The maxima in ql(k,) for 0.4 < kV(d/2n)< 0.6 is indicative of limited layer registration in the e, direction. If layers were perfectly registered such that each particle sat in a 3-fold hollow of adjacent layers, but the layers were only influenced by adjacent layers, the planes ofparticles would choose randomly between the two possible twin sites, rl and r2 in Figure 19.6333-36This results in random registered stacking (RRS) with strong maxima in ql(k,) at k , = 0.61(2n/d) and in q2(kv) at k , = 0 and 1.22(2n/d). While having general features of RRS structure, the weak nature of the maxima in ql(k,) and q2(kv) indicates a lack of strong registratin in the 2, direction. However, the strong intensity maxima seen at k, = 0 demonstrate a
Chen et al.
2826 Langmuir, Vol. 10,No. 8, 1994
-1
250
91:oi
o
$
1
'
I
l
l
A
L
3
m
50
2
0 4
loo]
00
500
1000
1500
2000
2500
Time (s)
Figure 14. Shear stress as a function of shear rate for 4 = 0.53. Open squares were gathered on the Couette geometry with a large gap. Other symbols refer to data gathered with a tapered plug. Open and closed circles indicate maximum and minimum stresses,respectively. Open and closed triangles indicatetime-averaged stressesdetermined from plateau stress time traces (see Figures 15 and 16) at short and long times after the shear rate has been stepped up from 500 s-l.
0 0
500
1000
1500
2000
2500
Time (s) Figure 15. Stress as a function oftime determined at y = 2224 s-l after stepping up from p = 500 s-l for 4 = 0.53. large degree of long-range orientational order within the planes of particles oriented in the &-C, plane. Little structural information has been derived from the polycrystalline scattering patterns seen at small but finite shear rates. Loss of distinct intensity maxima when k, = 0 and the retention of distinct but very diffuse intensity maxima when k, > 0 suggest that crystallites retain some preferred orientation with close-packed planes parallel to the rheometer walls. At elevated shear rates, the suspension becomes amorphous, showing little distortion in the first Debye-Scherer ring ask, is increased. At intermediate shear rates, the structure fluctuates between an amorphous and layered structure. We thus conclude that, with increasing stress, the suspensions under consideration evolve from a structure consisting of weakly registered hexagonally close packed planes a t rest to polycrystals where long-range orientational order is lost to a sliding layer structure where longrange orientational order is reestablished and finally to an amorphous structure at elevated shear rates. As long
Figure 16. Stress as a function of time determined at p = 3522 s-l after stepping up from p = 500 s-l for 4 = 0.53. as a critical shear rate is not exceeded, these transitions are reversible except for the transition from polycrystals to a single-crystal structure when the shear rate is set to zero. However, when a small amount of energy is imparted to the polycrystalline suspension in the form of low amplitude oscillations,the polycrystals relax to a structure with apparent long-range orientational order, suggesting that hexagonally close planes represent a low-energy configuration for the suspension's equilibrium structure. Random registered planes, sliding layers, and amorphous microstructures have been reported for numerous colloidal crystals undergoing hear.^-^,^^-^^ However, reports of steady-state polycrystalline microstructures evolving to sliding layers are rare. In studies36as detailed as the ones reported here carried out on a suspension of 146-nm particles at a volume fraction of 0.33, while random registered structures were observed at rest and slidinglayers at elevated shear rates, at low shear rates, the suspensions retained long-range orientational order and sustained steady deformation with a zigzag flow or slip zone mechanism. Similar microstructures have been reported on suspensions of particles with diameters in the 110-nm range5s6at lower volume fraction. The suspension of 146-nm particles showed two local stress maxima. Metastable viscositieswere observed for stresses just above the two local stress maxima. Microstructures could only be probed near the second stress minimum where discontinuous shear thinning under constant stress was associated with a transition from zigzag or slip zone to sliding layer flow mechanisms. In the second stress minimum, the suspension microstructure was polycrystalline. However, the polycrystalline microstructure was observed only in the stress minimum and, as a result, only over a shear rate range covering one decade as opposed to the four decades in shear rate observed for suspensions of 229-nm particles. While having distinct low shear rate microstructures, the rheologies of the suspensions of 146-and 229-nm particles are remarkably similar. For example, trSIGo,tydlGo,and t*/Go are independent of volume fraction within a given particle size and independent of size for the 146- and 229nm diameter particles explored in detail. The local stress maxima reported in Figures 6 and 7 mark the nucleation of the sliding layer phase. Up to t * , the elastic energy stored in the deforming polycrystalline sample increases as indicated by the recoverable strain. At t*,the stored elastic energy passes through a maximum.
Rheological and Microstructural Transitions
Langmuir, Vol. 10, No. 8, 1994 2827
Figure 17. SANS patterns for the suspension at 4 = 0.53 (a) 5 min, (b)22.5 min, and (c) 30 min after stepping from p = 500 s-l to p = 3000 s-l.
Figure 18. Optical photographs of the Couette shear cell as a function of time after a sample at 4 = 0.45 sheared at p = 0.01 for 3600 s is subjected t o oscillatory shear at a maximum strain of 0.2 at 20 Hz: (a, left) 0 min, (b, middle) 40 min, (c, right) 90 min.
We note that the recoverable strains at zyd and z* are, within experimental uncertainty, the same for both suspensions studied here and have values the same as
those measured for suspensions of similar particles reported in previous ~ o r k . ~ ? ~ ~ These results suggest that flow is initiated in dense
Chen et al.
2828 Langmuir, Vol. 10, No. 8,1994
flow behavior is not reversible on lowering the shear rate due to the irreversible degradation of the low shear structure or the prohibitive time scales required for the microstructures to rearrange. Our studies clearly show that reversible stress or shear rate induced microstructural transitions can result in local stress maxima. The transition from sliding layer to amorphous microstructures has seen considerable effort. Hoffman21and Boersma et al.25926 argue that shear thickening is the result of a transition from interparticle particle to viscous force control of the microstructure and predict p, should be a decreasing function of volume fraction. Ackerson and Clark predict shear melting is an increasing function of volume fraction but, in arguments similar to Hoffman and Boersma et al., attribute shear melting to a shearinduced instability which destroys sliding layer flow. Balancing electrostatic and viscous forces between pairs of particles at small separations, Boersma et al. argue pc can be predicted from
ee z
0 O
t"
0 0 O O 0 0 O O 0
(b) Figure 19. (a)Hexagonalclose packed layer of particles in the plane. r A and r~are particle position vectors in the plane. There are two possible positions for the next layer of particles in the Bo direction if the layers are to be registered. These are indicated by 1 and 2. (b) Scattering pattern resulting from illumination of planes of particles shown with orientation in (a)normal to the plane. Intensity maxima are defined as q&, &e, 2 ), = ( 2 ~ t / d2d(d2/3), , kv) and q 2 = (2n2/3/d,kv).
1.2
4
h
5
5
1.0
i? cd 5 0.8
pk
(d
v
where EOE, is the product of the permittivity of free space and the continuous-phase dielectric constant, J!O,I is the particle surface potential, vc is the viscosity of the continuous phase, and a is the particle radius. The Debye-Huckel parameter is found from [2nbe2/~o~&gT11'2, with nb being the number density of ions in solution (e.g., M), e the charge on a proton, and kBT the product of Boltzmann's constant and the absolute temperature. h is a characteristic surface to surface separation (Ma = 2[(4,/ 4)lI3 - 11). Boersma et al. assume a packing structure where the maximum packing fraction, &, is n/32/3 (=0.605). The maximum packing density for slidinglayers is set by the volume fraction where particles become trapped in cages and is -0.64. For the suspension of volume fraction 0.53, eq 1predicts p, = 6.3 x lo4s-l using @m = 0.605 and a surface potential of 77 mV (determined from osmotic pressures with the perturbation model of Voegtli and Zukoski40. While we see fluctuations in stress similar to those reported by Boersma et al., y, is severely overpredicted by their model. Accounting for a more accurate representation of the microstructure does not improve the prediction. Our high shear measurements were made with rheometer tools separated by 7 pm, a tool gap/particle size ratio much smaller than that used in other work. In studies where gaps have been varied between 5 and 20 pm, we find the critical shear rate increases with gap Thus, the p, reported here is a lower bound on that expected for larger rheometer tool gaps. Tests of the force balance hypothesis of Boersma et al. require studies of how yc varies with 4 which have not been carried out here. However, on the basis of the shear history dependence of microstructures and stresses observed in our studies, the definition of pc becomes problematic. Stress fluctuations shown in Figures 15 and 16 have been attributed to large cluster formation and degradation. The variations in scattering pattern suggest that cluster formation accompanies a destruction of long-range orientational order across much of the rheometer gap. Our rheological and scattering investigations suggest the instability resulting in cluster formation is not sharp and depends on the sample's shear history.
3
-Oe6 0.4 0.0
0.2
0.4
0.6
0.8
1.o
k,d/27~ Figure 20. Intensity of neutrons scattered at ql (kv),open symbols,and qz(kv), closed symbols,for 4 = 0.53 at shear rates of 0 s-' (0,e),40 s-l (A, A), 500 s-l (A, tus), and 1000 s-l(O). colloidal crystals when a critical elastic energy density has been stored in the suspension. For higher stresses, the stored elastic energy increases.to a second critical density where sliding layers are nucleated. The reversibility and hysteresis seen in the rheology and microstructures as well as the metastable viscosities seen for a range of stresses above z* suggest that the polycrystallinelsliding layer transition is fluctuation driven and thus may be discussed in terms of the language of equilibrium first-order phase transitions. The nature of nonequilibrium phase transitions is, however, quite complex and poorly u n d e r s t o ~ d . ~ As~aJresult, ~ ~ ~ ~analysis along these lines will not be pursued further here. Local maxima in shear stresslshear rate behavior are most commonly reported in flocculated suspensions and have long been attributed to microstructural transit i o n ~ . ~In~most -~~ flocculated suspensions,the anomalous
(37)Abduragimora, L. A.; Rebinder, P. A.; Serb Servina, N. N. Kolloidn. Zh. 1955, 17, 184. (38)Bartenev, G. M.; Povarova, Z. G. Kolloidn. Zh. 1966,28, 171. (39) Bartenev, G. M.; Ermilova, N. V. Kolloidn. Zh. 1969,31, 169. (40) Voegtli, L. P.; Zukoski,C. F. J . Colloid Interface Sci. 1991,141, 79.
Rheological and Microstructural Transitions
V. Conclusions Here we report on the microstructural and rheological properties of two suspensions of 229-nm-diameterchargestabilized particles suspended in KC1. Microstructural transitions from long-range orientational order at rest to polycrystals occur when the dynamic yield stress is exceeded. A sliding layer microstructure is nucleated when a stress greater than t* is exceeded, and for a range of stresses greater than z*, polycrystals and sliding layers coexist at steady state. At still higher shear rates, the long-range orientational order of the sliding layer microstructure is slowly degraded. These transitions are reversible to increases or decreases in shear rate or stress. Normalizing critical stresses with the suspension's elastic modulus measured on the samples having longrange orientational order at rest show remarkable similarity to those reported on particles of both similar and different size. This result confirms that GOis a characteristic stress scale for ordered suspensions. In addition, that microstructural transitions occur at fured values of t/Goindicates that these transitions are linked directly to equilibrium thermodynamic properties. However, the detailed mechanisms for rheological and microstructural transitions are currently poorly understood. We have observed anomalous flow behavior (i.e., local stress maxima as shear rate is increased) for suspensions of charge-stabilized particles at volume fractions above the equilibrium ordering volume fraction for latex particles with diameters of 34, 229, 238, 255, 458,and 550 nm in a volume fraction range of 0.18-0.60.8~30~32 For suspensions of particles with diameters greater than 200 nm, polycrystals are the preferred low shear rate, steady-state microstructure and anomalousflow behavior is associated with the nucleation of sliding layers. The ubiquitous nature of anomalous flow in ordered suspensions and the existence of a polycrystalline phase suggest that the behavior investigated here, at least at low shear rates where viscous contributions to stress and microstructure are of little importance, may simply reflect how samples with face-centered cubic ground states sustain deformation. In closing we note that Bridgman reports a
Langmuir, Vol. 10,No.8,1994 2829 phenomenon close to an anomalous flow behavior in the deformation of metals subjected to high pres~ures.~l Recently, there has been growing attention to pressure- and shear-induced amorphization in molecular ~ r y s t a l s . ~Static ~ - ~ ~pressure, grinding, and shock compression have been shown to drive molecular crystals into an amorphous state. Lower shear stresses than hydrostatic pressures are required for this process, and the amorphization can be reversible. Our observation of shear-induced polycrystalline phases in colloidal crystals suggests that this phenomenon may be a more common material response than previously thought. As the colloidal samples considered here are well above the particle density where ordering is observed, we have investigated rheological and microstructural properties under conditions equivalent to molecular samples subjected to high pressures. "he extreme pressure regions required to observe these phenomena in molecular crystals suggest that, if stress-induced microstructural transitions have the same physical origin, colloidal crystals may provide a unique opportunity to characterize crystal mechanics under high stresses and pressures.
Acknowledgment. We would like to thank J . Barker, Charlie Glinka, H. Hanley, and G. Straty for considerable help in obtaining the SANS results reported here and Jim Grey for his photographichelp. L.B.C. acknowledges the support of the NSF Center for Advanced Cement Based Materials, and M.K.C. acknowledges support of the US. Department of Energy through the Materials Research Laboratory at the University of Illinois. B.J.A. acknowledges support of NSF Grant No. DMR-9122589. (41)Bridgman, P.W. Proc. Am. Acad. Sci. 1937,17,387. (42)Mishima, 0.; Calved, L. D.; Whalley, E. Nature 1984,310,393. (43)Winters, R.R.; Hammock, W. S.Phys.Reu.Lett. 1992,68,3311. (44)Roy, R. C. Proc. Reg. Soc. 1923,A103, 690. (45) Kingma, K. J.; Russell, C. M.; Hemley, J.; Mao, H. K.; Veblen, D. R. Science 1993,666. (46)Clarke, D.R.; Kroll, M. C.;Kirchner, P. D.; Cook, R. F.; Hockey, B. J.Phys. Rev. Lett. 1988,60, 2156. (47)Minowa, K.;Somino, K. Phys. Reu. Lett. 1992,69, 320. (48)Christie, J. M.; Ardell, A. J. Geology 1974,2,405.
