Furuday Discuss. Chem. SOC., 1990,90, 91-106 Structure of a Binary Colloidal Suspension under Sheart Howard J. M. Hanley, John Pieper$ and Gerald C. Straty Thermophysics Division, National Institute of Standards and Technology, Boulder, CO 80303, USA Rex P. Hjelm Jr and Philip A. Seeger LANSCE, Physics Division Los Alamos National Laboratory, Los Alamos, NM 87545, USA Neutron scattering intensities from an aqueous mixture suspension of 91 nm polystyrene latex particles and 54nm silica particles are reported in the range 0.02 < Q/nm-' < 0.2, where Q is the momentum transfer. The sus- pension was dense at a mixture volume fraction of 0.15, and the poly- styrene/silica particle ratio was ca. 1.7 : 1. Results are given for the suspension at rest and under shear. The sheared data were obtained with a concentric cylinder shearing apparatus constructed and tested at the SANS facility of the National Institute of Standards and Technology and the pulsed neutron facility, LANSCE, of the Los Alamos National Laboratory.The design and operation of the cell is described. The shear-influenced behaviour of the mixture is compared with and contrasted to that of a pure polystyrene suspension that can form a crystal lattice in equilibrium, but which melts to a liquid-like structure under shear. A method is proposed to measure, by contrast matching or variation, the polystyrene and silica partial scattered intensities from the mixture suspension in H20-D20 solvents of different scattering-length densities. Estimates of the partial structure factors are given.Straty and co-workers',' have built an apparatus to investigate, by neutron scattering, a liquid under shear. The apparatus is versatile, rugged, designed to cover a wide range of shear rates and operating temperatures, and simple; the liquid is sheared between a stator and the rotating outer cylinder. We recently presented results on the microstructure (local particle order) of colloidal suspensions using the apparatus on the SANS (small-angle neutron scattering) facility at the National Institute of Standards and Technology (NIST)' and the pulsed neutron source of the Los Alamos Neutron Scattering Center (LANSCE).3 In this paper, data from a dense aqueous mixture of polystyrene latex spheres with approximate diameter (cps) 91 nm, and silica spheres with approximate diameter (csi) 54 nm are presented and compared and contrasted with the previous measurements.The species were selected deliberately to differ significantly in size. The mixture volume fraction was = 0.15 and the polystyrene-to-silica number density ratio was approximately 1.7 : 1. Several reasons make a mixture suspension an attractive candidate. First and foremost, pure colloidal suspensions have been studied extensively by light: X-rays,' and neutrons,' but not many authors have published on mixtures.6-'o The objective here was to investigate the microstructure of the mixture through the structure factor or the t Publication in part of NIST (formerly The National Bureau of Standards), not subject to copyright in $ Permanent address: Department of Physics, University of Colorado, Boulder, CO 80309, USA.the U.S.A. 9192 Binary Colloidal Suspension under Shear pair correlation function in the usual way. We wished, however, to distinguish the contribution of the polystyrene from that of the silica. To do this, the polystyrene or the silica are contrast matched3 from the mixture suspension in H20-D20 solvents of different scattering-length densities. The behaviour of a pure suspension or simple liquid under shear is very interesting. The phenomenon of shear-induced melting of a pure suspension is well documented by light4.” and neutron2312 scattering data, and the experiments are reinforced by computer-simulation results from model liquids.13-15 There has also been recent work on shear-induced ~rdering.’~~’’ By contrast, the behaviour of mixtures under shear is not known, and one of our objectives was to carry out some preliminary experiments.The paper is organized as follows. The basic equations for scattering from a colloidal suspension, and for the contrast matching from a mixture, are reviewed. The shearing apparatus is described and the sample preparation discussed. To set the stage for the mixture results, neutron scattered intensities from a pure suspension of the 91 nm polystyrene from the same batch sample used to make up the mixtures are presented first. We comment on the results in the context of the shear-induced melting phenomenon. The scattered intensities are then presented for the mixture at rest and under shear. Comparisons with the data from the pure sample are given. Estimates of the partial structure factor of the polystyrene and silica in the mixture are reported, and discussion remarks, conclusions and suggestions for further work complete the paper.Scattering Equations The basic expressions for the scattering from a suspension are outlined here for refer- ence.l8*l9 The scattered intensity ( I ) is proportional to the differential cross-section per unit volume of the sample, dC(Q)/da. Only the coherent contribution is of interest here. Q is the scattering wavevector for neutrons of wavelength A given by Q = ( 4 ~ / h ) sin e/2. ( 1 ) Here, 8 is the scattering angle, and Q = ko - k , ; ko and k , are the incident and scattered wavevectors, respectively, of the radiation; ko = 27r/h.If each colloidal particle is considered as a scattering centre, the expression for the cross-section is separated into terms for single particle and for interparticle particle scattering. For spherical particles where F ( Q ) is the form factor d w d f l = PIF(Q)I~S(Q) (2) F ( Q ) = C bjexp(iQ.Xj) and S ( Q ) is the structure factor, S(Q) = ( 1 ~ ~ ) (?? exp L ~ Q - ( ~ i - ~ i t ) ~ i i’ (3) (4) In eqn (2)-(4) Ri is the vector for the centre of mass of colloidal particle i, xj is the position of nucleus j relative to the centre of mass of particle i, p is the particle number density in volume V ( p = Np/ V ) , and bj is the scattering length of nucleus j . The measured intensity of interest is the intensity with respect to that of the medium of scattering length density, pm.In terms of pm, the form factor for a monodisperse system of spherical particles is where ps is the scattering-length density of the particle of volume radius R and volume, Vp = (4/3)nR3, and the function j ( Q R ) / ( Q R ) is F ( Q ) = V,(p,-p,)[3j(QR)/QR)I ( 5 ) j ( QR)/ QR) = (sin QR - QR cos Q R ) / ( Q R ) ~ . ( 6 )H. J. M. Hanrey et al. 93 The scattering-length densities are calculated from the operational definition where pmo' is the molecular density, and N ( j ) is the number of nuclei of type j per molecule of type k Thus, in terms of the particle volume fraction + = pV, we have Note that we can emphasize the role of the scattering-length density by writing eqn (8) as dC/da z Np(ps - pm)*I' (9) where I' contains all elements, other than those that depend directly on the scattering- length density, of the form factor and structure factor.In a typical experiment,20 the scattered intensity from the sample, I(sample), is measured at the volume fraction, +(sample), of interest. Then I(di1ute) is measured for a dilute sample at +(dilute) in the same apparatus under the same experimental conditions. The standard working expression to estimate the structure factor of a monodisperse single-component suspension follows S( Q ) = I(sample)c$(dilute)/[ I(dilute)+(sample)]. (10) Contrast Matching from Mixtures In the notation of eqn (9), the equation for a binary mixture can be written dx(Q)/damix z L C JNpJNqpp(mix)pq(mix)I~q; p7 = 132. (11) Pk(mix) = (pk - Pm); k = p7 4- (12) (13) dC(Q)ldap Np(pp - p m q ) I p p - (14) P 4 Here Suppose, therefore, pm is set equal to pq (designated pmq), eqn (11) becomes 2 1 Similarly, 2 1 dC( Q)/daq Nq(pq - Pmp) I q q * We have, therefore, eqn ( l l ) , (13) and (14) from which the total and the partial cross-sections can be extracted by appropriate manipulation of pm .Experimental Shearing Cell The apparatus is a Couette-type concentric cylinder fused quartz sample cell, coupled to a computer-controlled drive mechanism and a computer-controlled thermostat sys- Neutrons are scattered from the fluid in the annular gap between the cylinders (see fig. 1). A thermostatting fluid can be circulated within the inner stator, but is excluded from the neutron path by the sealed cross tube, see fig. 2. The outer cylinder is driven by a microprocessor-controlled d.c.servo motor, which is programmed for any desired combination of velocity, acceleration, delay time, direction of motion or rotation angle. We have verified that these variables are precise to within 0.1% for at least 24 h of continuous operation. A personal computer interfaces the motor and thermostatting controls with the operating and data acquisition system of the neutron facility, so that full automation of the experiment can be achieved if necessary.94 Binary Colloidal Suspension under Shear IB X Y Fig. 1. Schematic drawing of the shearing apparatus. M, drive motor; CR, cell rotor; CS, cell stator; TT, torque transducer; A, alignment coupling; X and Y, micrometer adjustment screws; Z, vertical adjustment. The neutron beam is directed to point B.The drive motor has an operating range from less than 0.006 r.p.m. to over 5400 r.p.m. delivering a maximum torque of ca. 5 N m. Drive system characteristics are shown in fig. 3. Shear rates of over 3.0 x lo4 Hz are possible with the present cell configuration; namely, the cell has a nominal mean diameter of 57 mm and an annular gap of 0.6 mm (1 r.p.m. is equivalent to a shear rate, G, of 4.89 Hz under these conditions). At present, the neutron beam is directed along a cell diameter if fluid is used as the thermostatting medium, but new cells are under construction that will allow the beam to be directed off centre. The gap width, and the range of attainable shear rates, will be adjustable by appropriate substitution of the stator and outer cell with others of different diameters.Sample Preparation Stock suspensions of a 30% mass fraction suspension of polystyrene in H20 and a 30% mass fraction suspension of silica (‘Ludox’?) in H 2 0 were available. According to the manufacturer’s specifications, ups = 91 nm and is ca. 7% polydisperse, and uSi = 50 nm and is essentially monodisperse. The estimates for polystyrene were confirmed by electron microscope imagery. Having scattering data, we now estimate the silica to be closer to 54nm in diameter. t The trade name is used to identify the product and does not imply any endorsement by NIST.H. J. M. Hanley et al. - sample region thermostatting fluid 5 E 4 z - 2 3 2- 2 2 1 Fig. 2. Details of the shearing cell. shear rate/ lo4 Hz 0.4 1.1 1.8 2.5 3.2 0 0 10 20 30 40 50 60 70 80 90 95 (1 200) (2400) (3600) (4800) rev.s-' (rev. min-') Fig. 3. Viscosity-torque relationship for the cell with its present configuration, see text. Two sets of sample solutions for the scattering experiments were prepared from the stocks; a polystyrene suspension, and mixtures of the polystyrene from the same polystyrene batch sample with silica in D20 and in D20-H20. Initial experiments with pure polystyrene were carried out in suspensions of volume fraction (4) 0.3, 0.085 and 0.06 in deionized water, the latter in a H20-D20 mixture. The suspensions were expected to be crystalline at these initial conditions based on the 4-ion concentration phase diagram determined from an X-ray study of the same batch sample.' To make these suspensions, the 30% polystyrene stock was deionized by tumbling with resin and diluted when necessary. The tumbling process was unsatisfactory in that the deionized 30% samples tended to aggregate over a period of days and had to be centrifuged or filtered before use.For later experiments, the polystyrene was96 Binary Colloidal Suspension under Shear outer cylinder 1 detector Y ,1z X Z Fig. 4. Neutron scattering geometry for the NIST SANS and LANSCE runs. Shear = du,/dy. purified by dialysis with an ion-exchange resin without agitation; which proved a more satisfactory procedure. For the mixture, the polystyrene solutions were subsequently dialysed with NaOH to replace the hydrogen counter-ions by Na+ to give a neutral pH in the polystyrene-silica suspensions.(The polystyrene-silica suspensions had to be pH neutral to ensure the silica component remained stable in suspension.) The silica was purified by filtration through a mixed-bed exchange resin and stabilized22 by adding NaOH solution until the pH reached ca. 8. Alternate solutions for each sample in D20 were prepared by dialysis against D20 in a closed container. Using measured values of the volume fractions of these treated samples, mixtures were made up for a mixture volume fraction +mix = 0.15 with ca. 64/36 polystyrene to silica number density ratio. The mixtures were pH 7-8 with an ion concentration of ca. 40 pmol ~ m - ~ . The mixture solvent was pure D20, or the appropriate D20-H20 proportion for contrast matching. A mixture with 28% D20 was prepared to match out the polystyrene, and a mixture with 60% D20 to match out the silica.The per cent D20 chosen corresponds to effective scattering-length densities (10'' cm-2) of 1.40 and 3.6 for the polystyrene and silica, given values of -0.56 and 6.4 for H20 and D20, respectively. Dilute solutions, 4 = 0.01, of pure polystyrene, pure silica and the mixture were prepared in D20. Scattering Measurements Scattering data were measured at the NIST and Los Alamos neutron facilities. Data were taken for the samples at rest and under shear with the scattering geometry of fig. 4 for all the experiments. In Cartesian coordinates the gradient, V, is in the y coordinate and the velocity, u, is along x. The shear-induced melting of the polystyrene suspensions was investigated at the NIST SANS facility.2 For these SANS measurements, a seven-beam converging pinhole collimation system23 illuminated a circle of ca.20mm diameter at the sample. The Q-range covered by the instrument's two-dimensional position-sensitive detector was 0.03-0.6 nm-I, given an incident wavelength of 1.2 nm, and a sample-to-detector distance of 3.6 m. Scattering intensities were recorded for the samples at rest with the cell stationary, and under shear with the outer cylinder rotating at speeds from 0.6 to 1500 r.p.m. The intensity data were corrected for the empty cell contribution, background scattering and sample transmission. We verified from previous that multiple scattering wouldH. J. M. Hanley et al. 97 ( C ) ( d ) Fig. 5. Scattered isointensity contours in the x-z plane from the 4 = 0.06 sample.( a ) from the system at rest; ( b ) from the system subjected to a shear rate G = 2.93 Hz; ( c ) G = 58.7 Hz; and ( d ) G = 176.0 Hz (see text). In the isointensity plots, the magnitude of Q ranges from Q = 0 at the centre of the plot to its maximum at the corner. be insignificant given the sample thickness here. The samples are strong scatterers, so the incoherent contribution to the scattering was taken to be negligible at the Q values of interest. The mixture was investigated initially3 on the time-of-flight (TOF) LQD instrument at LANSCE.25 The instrument was configured with a magnesium oxide filter to remove neutrons of wavelength shorter than ca. 0.15 nm from the beam.26 The moderator to sample distance and the sample to detector distance were 8.945 and 3.76 m, respectively. Data were collected into 147 TOF bins for wavelength determination. The width of each bin corresponded to A A / A = 1.6%.The raw data were corrected for the background and incoherent scattering of the medium and reduced to absolute units of differential scattering probability per unit solid angle as functions of Q by the methods described elsewhere.26327 The results are presented as differential cross-sections dC( Q ) / d n per unit volume in units of cm-'. Results Polystyrene We first display some of the results obtained for the pure polystyrene suspension from the NIST SANS experiments. Fig. 5 and 6, extracted from ref. (2) are the isointensity plots of the experimental scattering intensity in the x-z plane (fig.4) for the + = 0.06 and 0.30 samples, respectively. The sample was at rest in fig. 5 ( a ) . Six high-intensity spots are observed at Q == 0.05 nm-', with evidence of six less intense spots at Q == O.lOnm-'. As noted in ref. ( 2 ) , we interpret the inner grouping to arise from a two- dimensional hexagonally closed-packed layer with the close-packed direction along u. At this point we should state that we have no clear-cut way of knowing if the system at rest implies that the system is in equilibrium. Hess has observed28 that whether or not spots can be seen in the intensity pattern for an unsheared very dense suspension98 Binary Colloidal Suspension under Shear Fig. 6. Scattered isointensity contours in the x-z plane from the 4 = 0.30 sample; (a) from the system at rest; (b) from the system subjected to a shear rate of G = 58.7 Hz; ( c ) G = 293.4 Hz and ( d ) G = 1467.0 Hz (see text). depends on the time between the loading of the sample in the cell and the measurements.It is likely that a very dense unsheared colloidal suspension is polycrystalline. The mere act of introducing the suspension into a cell, or subjecting the system to a very low shear rate, defines a structure that may actually be that of the true equilibrium state. Fig. 5( 6) and (c) are typical of the intensity pattern from the sheared sample as the shear rate was increased. The suspension was subjected to shears of 2.93 Hz [fig. 5(b)], and 59 Hz [fig. 5(c)]. The inner six-fold symmetry is weakened with increasing shear and only four intensity maxima are seen, most probably because of a loss of correlation in the x-direction when the shearing motion forces one two-dimensional layer past another in the o direction.The intensity was liquid like at higher values of the shear, e.g. fig. 5 ( d ) for 176.0 Hz. The sequence was reversible and repeatable to within the resolution of the instrument. Isointensity contours for the 4 = 0.30 suspension are given in fig. 6. For the sus- pension at rest we surmise that the scattering pattern arises from the two-dimensional hexagonally close-packed layer along the o direction with little indication of structure in the gradient direction [fig. 6(a)]. Six-fold symmetry existed when the system was sheared, but the upper and lower spots became less intense with increasing shear.The crystal-like structure vanished, and the intensity pattern is that of a liquid [fig. 6(d)]. Polystyrene/Sitica Mixture Fig. 7(a) is the LANSCE LQD scattered intensity from pure polystyrene in D20 at rest, volume fraction # = 0.17. The pattern is consistent with fig. 5 and 6. Compare, however, with fig. 7(6), which is a typical pattern from the polystyrene-silica mixture in D20. The lattice-like order is not present. Possible structures in a mixture suspension are discussed by Lindsay and Chaikin6 who point out that a mixture structure may take considerable time to develop afterH. J. M. Hanley et al. 99 Fig, 7. (a) Typical scattered intensity from a dense pure polystyrene suspension, here at 4 = 0.17. The six intensity maxima are clear.The pattern is consistent with that from two-dimensional hexagonally closed-packed layers along the v direction [see fig. 5 and 6, and ref. (2)]. ( b ) Typical pattern from a mixture of polystyrene with silica. The intensity is liquid-like with no evidence of a lattice structure or anisotropy. All the mixture intensity patterns, for the mixture at rest and under shear, were of this general appearance. initial mixing of the components. Here, the time between sample preparation and the measurements was of the order of an hour. No special effort was made to agitate the mixture, or to allow the mixture to settle in the cell. The measurements for the sample at rest were made on the timescale of minutes. However, no observable lattice structure was induced when the mixture was subjected to a very low shear (much lower than would be expected to cause shear melting).Since we have argued that a very low shear may define any structure that is present for a pure suspension, it is very probable that the mixture does not have a definite lattice. Contrast Matching Fig. 8 gives the differential cross-sections versus Qa for pure polystyrene in D20 at a volume fraction 4 = 0.015, and for pure silica in D20 at # = 0.01 1. The heights of the possible maxima at Qa =r 0.04 are a function of the scattering-length densities and volume fraction difference and have been adjusted to be about equal. The diameter ups has been set at one for convenience. The dilute intensities were expected to scale with Qa.100 2500 2000 0" 1500 % .C Y m m 1000 500 Binary Colloidal Suspension under Shear 8 0 0 ra 8 9 0 0 0.05 0.1 0 0.1 5 0.20 Fig.8. Scattered intensities, expressed as the differential cross-section (,/cm-'), of dilute pure polystyrene and of dilute pure silica as a function of Qu with ups set to one. The diameter, asi, has been scaled to 0 . 6 5 ~ ~ ~ . 0, Polystyrene; 0, silica. 500 400 g 300 .- c) u 7 s b 200 100 0 0 0.05 0.10 0.1 5 0.20 Fig. 9. The differential cross-section from the 4 = 0.15 mixture (curve). U, The partial differential cross-sections of the polystyrene in the contrast-matched mixture. 0, The partial differential cross-sections of the silica in the contrast-matched mixture. Qu has been scaled as for the dilute pure samples. The positions of the peaks do not superimpose.H.J. M. Hanley et al. 3000 I 2000 h Y v) .- c i v 1000 x E E CI .I v) Y .- 0 -1 000 101 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 Qu Fig. 10. Estimate of the polystyrene-silica intensity evaluated from eqn (1 l), using eqn (12) and (13) for the mixture at equilibrium (O), and under shear (0) at G = 147 Hz. We have assumed u12 = ( U ~ ~ U , ~ ) ~ ' ~ . The curves for the mixture and for the sheared mixture are given by the and 0, respectively. 2500 2000 .- 2 1500 % Y u v1 8 1000 0 0 0.1 0.2 Qp Fig. 11. Effect of shear on the cross-section of the mixture. The intensity falls at the Qa corresponding to the peak, and the drop is essentially independent of the applied shear rate. The curve is the run at equilibrium; +, sheared mixture at G = 147 Hz; 0, the mixture at G = 5.870 Hz.102 Binary Colloidal Suspension under Shear 0 500 400 .- 300 Y u aJ 0 0.1 Qu 0.2 100 0 0 0.1 Qff 0.2 Fig.12. ( a ) The cross-section of silica in the mixture at equilibrium (+) and under shear (0) at G = 146 Hz. No difference is observed. ( b ) The cross-section of polystyrene in the mixture at equilibrium (+) and under shear (0) at 146 Hz. Compare with fig. 11. Superposition of the curves with the ratio of aP,/asi=91/54 was poor, but better if aps/asi = 1.54, a ratio that is not inconsistent with the estimated uncertainties of the polystyrene and silica diameters. Fig. 9 represents the contrast variation measurements. Three curves are given: the cross-section of the mixture in D20 and the partial cross-sections of the polystyrene in the 60% D20 solvent and the silica in the 28% D20 solvent.The heights of the peaks have been adjusted to be approximately equal and Qa has been scaled as for the dilute pure samples, with aiix = (1.7a$ + azi)/2.7 and ups set at one. The polystyrene-poly- styrene and the silica-silica contributions to eqn (1 1) were calculated from the appropri- ate contrast-matched data, and the polystyrene-silica cross intensity estimated by sub-H. J, M. Hanley et al. 103 traction from the total mixture intensity (see fig. 10). Comments on these curves are deferred until the discussion. Scattering from the Sheared Mixture Although the definite shear-induced structures observed in sheared pure polystyrene suspensions were not seen in the polystyrene-silica mixtures, the maximum intensity was found to fall significantly when a shear was applied (fig.11). Furthermore, the drop was essentially independent of the applied shear rate to within the resolution of our instruments. Measuring the intensity from the contrast-matched sheared mixtures indicates that the drop comes exclusively from the polystyrene contribution (fig. 12). The cross-term for the sheared mixture was evaluated from eqn (11) and plotted in fig. 10. Remarks and Conclusion Neutron-scattered intensities were measured from a dense polystyrene-silica aqueous suspension at rest and under shear. Much of the work is preliminary but we have two definite, if at this stage qualitative, results. First, we were able to distinguish the scattered intensity from each of the two components by contrast matching selectively the com- ponents with an appropriately mixed H20-D20 solvent.Contrast matching or variation has been used to separate partial structure factors in biological and, using isotope substitution, is a well known technique to study the structure of alloy^.^' However, we have applied it to a liquid mixture. Secondly, the behaviour of the mixture under shear is definitely different from that of a pure suspension under shear. The scattered intensities from the mixture were compared with those from a pure polystyrene suspension that had a lattice structure at rest. The pure suspension lattice melted to a liquid structure under shear. We observed, however, that the polystyrene-silica mixture did not have a defined lattice at rest (see fig.7). Consequently, the mixture obviously did not display the phenomenon of shear-induced melting, but the scattered intensity fell (in the Q range investigated) when the mixture was sheared. Clearly, the shear- induced behaviour of mixtures requires further experimental work. A satisfactory conclusion to this study would be to estimate the structure of the mixture from the scattered intensities. We can write eqn (1 1) in terms of the Ashcroft- Langreth partial structure factors32 and assume that an eqn (10) is valid for each term. Thus, in principle the partial structure factor of species k is evaluated from eqn (10) where I(samp1e) is the contrast-matched intensity for k and I(di1ute) is the intensity of the form factor of pure k. For technical reasons, we were not satisfied that the dilute runs of fig.8 were of sufficient precision to be regarded as legitimate form-factor data. Very recently,33 however, some of the equilibrium LANSCE measurements were repeated on the same samples using the D11 spectrometer of the ILL, Grenoble. For these experiments, the mixtures were loaded into a standard quartz cell of pathlength 1 mm. The sample-detector distance was 20 m and the neutron wavelength was 1.2 nm giving an experimental Q range of 0.01 < Q/nm-' < 0.12. were placed on an absolute scale with respect to the water standard measured at 5 m. Corrections for the empty cell contribution and background scattering were made, but not for the contribution of the water solvent or incoherent scattering. The D11 data and the Los Alamos data were consistent to within the resolution of the instruments for the Q range of overlap.There are still some doubts that the dilute runs represented the form factors correctly; nevertheless, fig. 13 is an estimate of the polystyrene-polystyrene and silica-silica partial structure factors in the mixture. The apparent structure factor for polystyrene is con- sistent with other estimates from X-ray data' and our independent SANS from The ILL104 o o o o a " ~ Binary Colloidal Suspension under Shear 1 I@ l3a a a a ifl, Y , , , , , , I , ; , 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 r / u Fig. 14. The Fourier transforms of the estimated partial structure factors (polystyrene, 0; silica, 0) shown in fig. 13, plotted versus r / a with ap,=91 nm and asi=61 nm.The position of the silica peaks depends on the value selected for asi, which is uncertain to 10%. the same batch sample of the pure polystyrene. Fig. 14 shows the Fourier transforms of the polystyrene and silica curves. Since g ( r * ) = 1 +- I [ S ( W - 1 I ( W sin (Qar*) d ( Q 4 (15) 127r+r* where r* = r / a , these transforms are presented as approximations to the radial distribu- tion functions for polystyrene and silica in the mixture at rest. They suggest that theH. J. M. Hanley et al. 105 polystyrene structure at = 0.15 is very close to the structure of pure polystyrene at an equivalent volume fraction with a main peak in g ( r ) at ca. 137 nm, but that the silica has a weaker lattice with a peak at ca. 300nm. The numbers make sense but have to be regarded with caution.Furthermore, only the results from a system with one particular mixture volume fraction and one particular polystyrene-to-silica particle ratio are avail- able, so any general conclusions on the structure of a mixture are not possible at this time. Clearly the structure of the mixture will depend on factors such as the volume fraction, the particle size ratio, the number density ratio and the surface-charge densities. Overall, the contrast variation technique applied to suspension mixtures is satisfactory and has led to results that encourage further work. The structure data of the equilibrium suspension were not unexpected and were anti~ipated~~ by simulation of model l i q ~ i d s ~ ~ , ~ ~ and by simple theories of although the apparent long-range structure of the silica in the mixture was surprisingly evident.However, the way in which the mixture behaves under shear is new and will be investigated in more detail. The work of H. Hanley and G. C. Straty was supported in part by the U.S. Department of Energy, Office of Basic Energy Sciences. J. Pieper acknowledges a National Science Foundation Graduate Fellowship. The LQD of LANSCE is supported by the US Department of Energy and under contract W-7005-ENG-32 to the University of Califor- nia. We are very grateful to C. G. Glinka of NIST and P. Lindner of the ILL for their contribution to this work. References 1 G. C. Straty, NISTJ. Res., 1989, 94, 259. 2 G. C. Straty, H. J. M. Hanley and C. J. Glinka, J. Stat. Phys., 1990, in press.3 G. C. Straty, J. Pieper and H. J. M. Hanley, Mol. Phys., 1990, in press. 4 For example, see B. J. Ackerson and N. A. Clark, Physica, 1983, 118a, 221, and references therein. 5 E. B. Sirota, H. D. Ou-Yang, S. K. Sinha, P. M. Chaikin, J. D. Axe and Y. Fujii, Phys. Rev. Lett., 1989, 6 H. M. Lindsay and P. M. Chaikin, J. Chem. Phys., 1982, 76, 3774. 7 S. Yoshimura and S. Hachisu, Prog. Colloid Polym. Sci., 1983, 68, 59. 8 W-H. Shih and D. Stroud, J. Chem. Phys., 1984, 80, 4429. 9 W. Hart1 and H. Versmold, J. Chem. Phys., 80, 1387; B. V. R. Tata, Y. R. Kesavamoorthy and A. K. 62, 1524, and references therein. Sood, Mol. Phys., 1987, 61, 943. 10 T. Okubo, J. Chem. Phys., 1987, 87, 5528. 11 R. L. Hoffman, Trans. SOC. Rheol., 1972, 16, 155. 12 B. J. Ackerson, J. B. Hayter, N.A. Clark and L. Cotter, J. Chem. Phys., 1986, 84, 2344; S. Ashdown, I. Markovic, R. H. Ottewill, P. Lindner, R. C. Oberthur and A. R. Rennie, Langmuir, 1990, 6, 303. 13 H. J. M. Hanley, Lectures on Thermodynamics and Statistical Mechanics, ed. A. E. Gonzales and C. Varea (World Scientific Press, 1988), p. 109. 14 H. J. M. Hanley, J. C. Rainwater, N. A. Clark and B. J. Ackerson, J. Chem. Phys., 1983, 79, 4448. 15 T. Weider, M. L. Glasser and H. J. M. Hanley, 1990, in press. 16 B. J. Ackerson and P. N. Pusey, Phys. Rev. Lett., 1989, 61, 1033; B. J. Ackerson, C. G. De Kruif, 17 S. J. Johnson, C. G. De Kruiff and R. P. May, J. Chem. Phys., 1988, 89, 5909. 18 S-H. Chen and T-S. Lin, in Methods in Experimental Physics, Vol. 23, Part B, Neutron Scattering 19 R.H. Ottewill, in Colloidal Dispersions (The Royal SOC. Chem. Spec. Pub. no. 43, London, 1982). 20 For example D. J. Cebula, J. W. Goodwin, G. C. Jeffrey, R. H. Ottewill, A. Parentich and R. A. Richardson, Faraday Discuss. Chem. SOC., 1983, 76, 37. 21 P. Lindner and R. C. Oberthur, Reu. Phys. Appl., 1984, 19, 759. 22 J. D. F. Ramsay and B. 0. Booth, J. Chem. SOC., Faraday Trans. 1, 1983, 79, 173. 23 C. J. Glinka, J. M. Rowe and J. G. LaRock, J. Appl. Crystallogr., 1986, 19, 424. 24 C. J. Glinka, D. A. Aastuen, H. J. M. Hanley and G. C. Straty, unpublished data available from the 25 P. A. Seeger, R. P. Hjelm and M. Nutter, Mol. Cryst. Liq. Cryst., 1990, 180A, 101. 26 R. P. Hjelm, J. Appl. Crystallogr., 1988, 21, 618. 27 R. P. Hjelm and P. A. Seeger, Physics Conference Series, 97, 367 (IOP Publications, Bristol, UK, 1990). N. J. Wagner and W. B. Russel, J. Chem. Phys., 1989, 90, 3250. (Academic Press, London, 1987). Thermophysics Division, NIST, Boulder, CO.106 Binary Colloidal Suspension under Shear 28 S. Hess, personal communication, Tech. Univer., Berlin, 1990. 29 P-J. Derian, L. Belloni and M. Drifford, Europhys. Lett., 1988, 7 , 243. 30 H. B. Stuhrmann, Neutron Scattering, ed. W. Glasser (North-Holland, New York, 1988), p. 444. 31 For example, J. E. Enderby, D. M. North and P. A. Egelstaff, Philos. Mag., 1966, 14,961; S . Eisenberg, J-F. Jal, J. Dupuy, P. Chieux and W. Knoll, Philos. Mag. A , 1982, 46, 195; H. Ruppersberg, Phys. Chem. Liq., 1987, 17, 73. 32 N. W. Ashcroft and D. C. Langreth, Phys. Rev., 1967, 155, 682; 1967, 159, 500. 33 H. J. M. Hanley, G. C. Straty and P. Lindner, Physica, 1990, in press. 34 W. Schmaltz, T. Springer, J. Schelten and K. Ibel, J. Appl. Crystallogr., 1974, 7 , 96; K. Ibel, J. Appl. 35 I. Markovic and R. H. Ottewill, Colloid Polym. Sci., 1986, 264, 65; 454. 36 D. J. Evans and H. J. M. Hanley, Phys. Rev. A , 1979, 20, 1648. 37 M. Huber and J. F. Ely, NIST Tech. Note no. 1331, 1989. 38 D. Henderson and P. J. Leonard, in Physical Chemistry, An Advanced Treatise, Vol. viiiB/ Liquid State Crystallogr., 1976, 9, 296. (Academic Press, New York, 1971), chap. 7, p. 413. Paper 0/02323J; Received 17th May, 1990