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Langmuir 1998, 14, 7364-7370
An Aggregation Index for Monitoring the State of the Suspensions Manjunath Subbanna, Sandhya Kokil, P. C. Kapur, and Pradip* Tata Research Development and Design Centre, Pune-411 013, India
Subhas G. Malghan National Institute of Standards and Technology, Gaithersburg, Maryland 20899 Received June 24, 1998. In Final Form: September 18, 1998 The state of aggregation in alumina, zirconia, and mixed alumina-zirconia suspensions is determined by particle size analysis of particles/aggregates in suspensions. An aggregation index is derived using principal component analysis of the size distribution data. The first principal component is shown to be an adequate single-value measure of the extent of aggregation in the suspension. Aggregation indices are obtained for suspensions prepared at different pH with and without the addition of different polymeric surfactants of varying dosage and molecular weight and employed for analyzing the state of the suspensions.
Introduction Depending upon the magnitudes of attractive and repulsive forces operating between particles,1,2 a suspension may range from a highly aggregated (coagulated or flocculated) to a fully dispersed system. The state of the suspension, in turn, impacts many macroscopic properties including stability, settling rate, sediment volume and compressibility, light transmission and scattering, viscosity, shear stress, and other rheological properties. A specific example is colloidal processing of precursor powders for fabrication of improved ceramic components.3-6 For efficient sintering and optimized properties of the sintered body, it is necessary to start with a homogeneous green compact, which is achieved only if the slurry is well dispersed.7-9 For reasons to be discussed subsequently, the problem of obtaining a well-dispersed slurry becomes considerably more complex in the case of mixed suspensions that are employed for the fabrication of ceramic composites, for example, alumina-zirconia bodies.10,11 Dispersion of powders is frequently accomplished by addition of suitable inorganic or organic reagents or polymeric surfactants. The most common mechanism by which powders in suspension are dispersed is through electrostatic and electrosteric repulsion, in which the surface charge characteristics of the powders are critical.12-15 * Corresponding author. (1) Horn, R. G. J. Am. Ceram. Soc. 1990, 73, 1117. (2) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: New York, 1992; pp 139-337. (3) Barringer, E. A.; Bowen, H. K. J. Am. Ceram. Soc. 1982, 65, C-199. (4) Kendall, K. Powder Technol. 1989, 58, 151. (5) Rice, R. W. AIChE J. 1990, 36, 481. (6) Leong, Y. K. Mater. Des. 1994, 15, 141. (7) Lange, F. F. J. Am. Ceram. Soc. 1989, 72, 3. (8) Yeh, T.-S.; Sacks, M. D. J. Am. Ceram. Soc. 1988, 71, 841. (9) Kimura, T.; Matsuda, Y.; Oda, M.; Yamaguchi, T. Ceram. Int. 1987, 13, 27. (10) DeLiso, E. M.; Rijswijk, W. V.; Cannon, W. R. Colloids Surf. 1991, 53, 383. (11) Bleier, A.; Westmoreland, C. G. J. Am. Ceram. Soc. 1991, 74, 3100. (12) Rao, A. S. Ceram. Int. 1987, 13, 233. (13) Malghan, S. G.; Premachandran R. S.; Pei, P. T. Powder Technol. 1994, 79, 43. (14) Premachandran, R. S.; Malghan, S. G. Powder Technol. 1994, 79, 53.
Evidently, the characterization of the state of dispersion, that is, the extent or degree of dispersion, is of considerable research interest and great technological importance. The more common techniques employed for this purpose include settling tests16 and rheological studies such as viscosity and yield stress measurements.17,18 A more direct manifestation of the state of the suspension, however, is the size distribution of suspended particles and aggregates.15 A variety of methods such as microscopy, photography, photosedimentation, and light scattering are employed for determining the size of aggregates in suspension. Farrow and Warren19 have reviewed these techniques along with their merits and demerits. Irrespective of whether it is available in the form of a table, a fitted mathematical function, or a graphed curve, the size distribution, unfortunately, is not exactly a convenient quantitative measure of aggregation.20 Ideally, a reasonably accurate single-parameter description of the size distribution is needed for the purpose of interpretation of data, modeling of the colloidal process and for analysis and comparison of suspensions generated under different processing conditions. The commonly used single parameter characteristic size, such as mean, median, or some other percentile size,21,22 is ordinarily neither unique to a distributionsin fact, a literally infinite number of distributions can exhibit a given characteristic sizesnor adequately representative of it. The problem is compounded by the fact that the measured aggregate size distributions may turn out to be bimodal under some conditions, and are unlikely to be represented meaningfully by a conventional one-parameter size index. It is well-known that a distribution can be described to any degree of accuracy by its percentiles. Whereas, three (15) Pradip; Premachandran, R. S.; Malghan, S. G. Bull. Mater. Sci. 1994, 17, 911. (16) Ramakrishnan, V.; Pradip; Malghan, S. G. Colloids Surf. 1998, 133, 135. (17) Ramakrishnan, V.; Pradip; Malghan, S. G. J. Am. Ceram. Soc. 1996, 79, 2567. (18) Manjunath, S.; Pradip; Malghan, S. G. Chem. Eng. Sci. 1998, 53, 3073. (19) Farrow, J.; Warren, L. In Coagulation and Flocculation; Dobias, B., Ed.; Marcel Dekker: New York, 1993; pp 391-426. (20) Kelso, J. F.; Ferrazzoli, T. A. J. Am. Ceram. Soc. 1989, 72, 625. (21) Bagchi, P.; Vold, R. D. J. Colloid Interface Sci.1975, 53, 194. (22) Koglin, B. Powder Technol. 1977, 17, 219.
10.1021/la980744e CCC: $15.00 © 1998 American Chemical Society Published on Web 11/26/1998
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for monitoring the state of the suspension greatly facilitates the evaluation and comparison of suspensions prepared under different processing conditions, as discussed subsequently in this paper. Principal Component Analysis of Aggregate Distributions
Figure 1. Outline of the proposed aggregation index based on principal component analysis of the aggregate size distribution.
parameters such as 25-, 50-, and 75-percentile may provide only a crude approximation of the original distribution, a more satisfactory description can be obtained by employing more parameters. The main difficulty with this approach is that the large number of distribution percentiles invariably results in a rather unwieldy and cumbersome analysis, which can lead to ambiguous results of limited utility. The problem can be overcome by employing a linear combination of the percentiles as a representative single-value measure of the distribution. It turns out that the principal component analysis provides a formal statistical framework for carrying out this transformation. Instead of using conventional indices such as a mean size, surface mean size, volume mean size, average of 25th and 75th percentile, etc., we have found that the first principal component of the distributions, which is also a single number, is a more representative measure of the distributions. We have thus introduced a new aggregation index for a quantitative assessment of the state of the suspension. As shown in Figure 1, the proposed index is in fact the first principal component in the vector of principal components obtained by the transformation of the vector of percentiles of a size distribution of aggregates. It can be used either for qualitative assessment of the suspension or for establishing quantitative relationship with input, process, and equipment variables and parameters in processing of a suspension. The index has been tested extensively on alumina, zirconia, and mixed alumina-zirconia suspensions in which dispersionaggregation was manipulated by pH control and or by addition of polymeric surfactants. These suspensions are used in our ongoing research on colloidal processing of mixed slurries for the fabrication of ceramic composites of alumina and zirconia. Ensuring complete dispersion and homogeneous mixing of the oxides, in the face of the heterocoagulation between the constituent powders,15-18,23-26 is of utmost importance in this fabrication technique. Availability of an aggregation index as a tool (23) Hogg, R.; Healy, T. W.; Fuerstenau, D. W. Trans. Faraday Soc. 1966, 62 [522, Part 6], 1638. (24) Kihira, H.; Matijevic, E. Langmuir 1992, 8, 2855. (25) Deliso, E. M.; Cannon, W. R.; Rao, A. S. Mater. Res. Soc. Symp. Proc. 1986, 60, 43. (26) Deliso, E. M.; Cannon, W. R.; Rao, A. S. In Advances in Ceramics; Somiya, M., Yamamoto, N., Yanagida, H., Eds.; American Ceramic Society: Westerville, OH, 1988; Vol. 24, p 335.
Details of the principal component analysis are readily available in many books on multivariate statistics and data analysis.27-29 Here we summarize only the essential steps involved in transformation of the original size distribution data into the first principal component. We begin with N measured aggregate size distributions, known as observations or responses. Each distribution is broken into P number of percentiles over a suitable range, for example, 5-percentile to 95-percentile. There is no restriction on N, the number of observations; in particular, it can be less or more than P. In the resulting N × P data matrix M, the element mij stands for Xj percentile (j ) 1, 2, ..., P) of ith distribution (i ) 1, 2, ..., N). The P × P dispersion matrix S, also known as the variance-covariance matrix, is given by
S ) M′‚M
(1)
where M′ is the transpose of M. Next, the 1 × P vector evl of eigenvalues and the P × P matrix Evc of eigenvectors of the dispersion matrix may be denoted by
evl ) Eigenvalues(S) ) [evl1, evl2, ...evlP-1, evlP]
(2)
and
Evc ) Eigenvectors(S) ) [Evc1, Evc2, ..., EvcP-1, EvcP]
(3)
where Evcj, jth column of the Evc matrix, is the jth eigenvector. The N × P matrix of principal components is given by
PC ) M‚Evc ) [pc1, pc2, ..., pcP-1, pcP]
(4)
The first column of the PC matrix, in particular, is the all-important first principal component, pc1, which can be derived from the following relationship also:
[]
pc1 ) M‚Evc1 pc11 pc12 . ) . pc1N-1 pc1N
(5)
In general, the remaining principal components are of no interest as these carry only a very limited amount of information on the aggregate distributions. For comparison with original distributions, we can reconstruct (27) Kendall, M. K. Multivariate Analysis; Charles Griffin & Co. Ltd.: London, 1980; Chapter 2, pp 13-29. (28) Cooper, R. A.; Weekes, A. J. Data, Model and Statistical Analysis; Heritage Publishers: New Delhi, 1983; Chapter 10, pp 240-277. (29) Llinas, J. R.; Ruiz, J. M. In Computer Aids to Chemistry; Vernin, G., Chanon, M., Eds.; Ellis Harwood Ltd.: 1986; Chapter V, pp 200239.
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Table 1. Specific Surface Area, Particle Size and Isoelectric Point of Alumina and Zirconia Powders
powder
source
alumina (A-16) Alcoa, USA zirconia (Unitec) Unitec, U.K. zirconia (SY 5.2) Z.Tech, Australia
isospecific surface particle electric point size area (m2/g)a (d50, µm)b (pH)c 9.3 3.32 14.9
0.426 0.735 0.261
7.6 6.0 6.65
a Measured using Flowsorb II 2300 BET surface area analyzer (Micromeritics, USA). b Analyzed on LA-910 laser light scattering analyzer (HORIBA Ltd., Japan). Resolution is 0.02 µm. c Determined from electrophoretic mobility data measured using Zeta Meter 3.0 (Zeta Meter Inc., USA).
Figure 2. Cumulative eigenvalues for principal components from all the size distribution data of the suspensions.
the N × P matrix of percentile data from the somewhat truncated information contained in the first principal component by the following relationship:
Mpc ) pc1‚Evc1′
(6)
Note that the dimensions of pc1 and Evc1′ are N × 1 and 1 × P, respectively, resulting in a N × P matrix Mpc. Depending upon how close the back calculated matrix Mpcis, to the original matrix M, the elements of pc1 can provide a moderately reasonable to highly accurate oneparameter description of aggregate distributions. It will be seen from eqns 4 and 5 that the principal components are linear combinations of the percentiles. This transformation entails a rotation of coordinate axes to a new coordinate system in which the principal components are orthogonal and uncorrelated. Therefore, the dispersion matrix of the PC matrix has only diagonal nonzero elements; all off-diagonal elements for covariance are zero. The coefficients of the linear transformation are given by the eigenvectors. Since the sum of the squares of coefficients in each eigenvector adds up to unity, the principal components may be perceived as a kind of weighted sum of the percentiles of a distribution. An important consequence of the transformation is the redistribution of the variance. Although the sum of the variance of P columns in the original data matrix is equal to the sum of the variance of P columns of the principal components, in latter case the first principal component has by far the greatest variance and the last principal component the least. This is duly reflected in the vector of eigenvalues, evl, whose elements are simply the variance of the principal components. An important implication of this redistribution and ordering of variance is that the first principal component carries the maximum amount of information about the distribution and the last has little or none. Experience shows that in aggregate size distributions, the first principal component carries nearly all the information, and as such, it is a valid and meaningful one-parameter measure of the aggregate distribution.
Figure 3. Comparison of the original aggregate size distributions of alumina, zirconia, and mixed alumina-zirconia (1:1) suspensions at different pH without surfactant, with those back calculated from the first principal component (P.C.).
is represented in parts per million (ppm), that is, milligrams of polymer per liter of solution. Suspensions having five percent solids by weight were prepared in deionized water by ultrasonic dispersion for about 5 min. For mixed suspensions, alumina and zirconia were taken in 1:1 volume ratio. Adjustment of the suspension pH was made with analytical grade nitric acid and sodium hydroxide solutions of 0.1 N concentration. Samples from the suspension were drawn and analyzed in the HORIBA LA-910, a laser scattering particle size analyzer. The pH in the sample cell of the analyzer was adjusted to match the suspension pH, prior to introducing the sample for analysis. The time interval between the sample preparation and size analysis was 10 min. Aggregation Index of Suspensions
Experimental Details The powders used in this study were alumina (A16), Zirconia (Unitec) and Zirconia (SY 5.2). The properties of the powders are given in Table 1. Three polyelectrolytic surfactants were used: (1) Darvan-C, ammonium polymethacrylate of molecular weight 15 000; (2) poly(acrylic acid) of molecular weights 2000 and 10 000, anionic polymers from Vanderbilt Co., U.S.A.; (3) Betz 1190, quaternized polyamine epoxy chlorohydrin of molecular weight 12 000, a cationic polymer supplied by Betz Co., U.S.A. The dosage of polymer in suspension
In all, 210 suspensions of the alumina, zirconia, and mixed alumina-zirconia were measured for size distribution of aggregates. These suspensions differed in pH and or surfactant additions. Only one measurement per suspension sample was needed to ascertain the distribution. Each distribution was broken into 15 percentiles corresponding to 1, 3, 5, 7, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, and 95 cumulative percent passing. Cumulative eigenvalues, of the resulting 210 × 15 data matrix, for all the size distribution data acquired, normalized to 100, are plotted in Figure 2 as a function of the number of
An Aggregation Index for Suspensions
Figure 4. Comparison of the original aggregate size distributions of alumina, zirconia, and mixed alumina-zirconia (1:1) suspensions at different pH with addition of 20 ppm PAA 2K, with those back calculated from the first principal component (P.C.).
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Figure 6. Aggregation index of alumina, zirconia, and mixed alumina-zirconia (1:1) suspensions.
Figure 5. Aggregation index of zirconia suspensions as a function of pH.
principal components. It will be seen that the first principal component accounts for more than 98% of the total variance. Accordingly, the first principal component is adequate for a realistic and accurate representation of the aggregate size data. This conclusion is confirmed in Figures 3 and 4 where a few representative measured aggregate size distributions are compared with those back calculated from the first principal component using eq 6. The former figure includes only the pH effect while the latter pertains to suspensions with 20 ppm of a poly(acrylic acid), molecular weight 2000 (PAA, MW 2K) surfactant. On the whole, the agreement is remarkably good. The relatively small discrepancies, especially in fine size range, could be due to a variety of reasons. One, while 98% of variance is accounted for by the first principal component, the remaining information is lost in the 14 principal components that have been discarded. Two, the nature of agreement between the original and reconstructed size distributions depends to some extent on the choice of the percentiles. A somewhat better fit may be possible in a
Figure 7. Size distributions of aggregates in fully coagulated alumina, zirconia, and mixed alumina-zirconia (1:1) suspensions
region of interest by selecting more percentiles in that range. Three, there is bound to be some noise in the experimental data which is unlikely to be distributed uniformly over the full size range. This in turn may bias the degree and nature of agreement somewhat. Despite these issues, it is reasonable to conclude that the first principal component provides an accurate index of aggregation and a convenient quantitative measure of the state of the suspension. Note that the aggregation index has a dimension of length and its value is of the same order of magnitude as the size range of the original distribution. Figure 5 tracks the aggregation index of zirconia suspensions in a pH range of about 3 to 11. As anticipated, the index attains its maximum value greater than 16 µm in the close vicinity of the isoelectric point of zirconia, falling to less than 2 µm at the extremes of the range. When subjected to ultrasonic treatment just prior to the
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Figure 9. Effect of dosage of poly(acrylic acid), molecular weight 2000 (PAA 2K) on the maximum aggregation index of alumina, zirconia, and mixed alumina-zirconia (1:1) suspensions.
Figure 8. Aggregation index of (a) alumina, (b) zirconia, and (c) mixed alumina-zirconia suspensions with poly(acrylic acid), molecular weight 2000 (PAA 2K)
size analysis, the aggregation index remains flat at less than 1 µm, irrespective of the suspension pH. The aggregation index curves of alumina, zirconia, and mixed alumina-zirconia (1:1 by volume) suspensions as functions of pH are compared in Figure 6. It is interesting to note that the maximum aggregation indices, associated with the coagulated state of the suspensions, turn out to be comparable in magnitude. Figure 7 shows that the size distributions at corresponding pH are also quite close to each other, as these should be in view of similar aggregation indices. This observation is consistent with the well-known DLVO theory which predicts maximum aggregation at the isoelectric point where repulsive electrostatic forces are nonexistent.15-17,23 As for the position of the maximum aggregation index of the mixed
suspensions the data are not sufficient to make a conclusive statement about the physics of mixed suspensions. It is at times convenient for comparison to normalize the aggregation index by the maximum aggregation index of the powder at the isoelectric point, without the addition of any other surfactant or aggregation/dispersion agent. Figure 8a shows four dimensionless normalized aggregation index curves for alumina suspensions containing 0, 10, 20, and 50 ppm PAA, MW 2K. The maximum index with 50 ppm surfactant is two and a half times higher than the standard and shifted by a pH of nearly 4.5 as compared to suspensions without any surfactant. Similar results are shown in Figure 8b for the aggregation index of zirconia suspensions as a function of pH with and without the surfactant addition. The observed trends are similar to those observed in the case of alumina suspensions. The maximum index with a surfactant addition of 50 ppm occurs at a pH of around 2.5. Figure 8c shows the normalized aggregation index for mixed alumina-zirconia suspensions with and without the surfactant addition. The maximum index at 50 ppm surfactant addition is observed at a pH of 4. The index in this case is nearly 3 times that of suspensions without any surfactant. It is interesting to note here that according to Leong6 the yield stress as a function of pH of concentrated zirconia suspensions exhibits similar behavior with the addition of PAA. The maximum value of yield stress not only shifts to the left on the pH scale but also the maximum yield stress decreases with increasing concentration of surfactant. However in the case of surfactants such as sodium dodecylamine sulfate (SDS) and dodecylamine hydrochloride (DAL), the maximum yield stress value increases with increasing surfactant concentration. The observations have been explained qualitatively in terms of electrostatic, van der Waals, steric, bridging, and hydrophobic forces. It is quite likely that our observations of aggregation index and the yield stress data of Leong stem from the same underlying interparticle interactions in suspensions. Figure 9 shows the effect of dosage of polymer PAA MW 2K on the maximum aggregation index of alumina,
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Figure 10. Effect of polymer dosage and molecular weight on the maximum aggregation index of alumina suspensions.
Figure 12. . Electrokinetic sonic amplitude (ESA) analysis and aggregation index of zirconia suspensions with BETZ 1190.
Figure 11. Electrokinetic sonic amplitude (ESA) analysis and aggregation index of alumina suspensions with Darvan C.
zirconia, and mixed alumina-zirconia suspensions. In all three cases, the index initially increases steeply and then progressively tapers off beyond the dosage of 20-40 ppm. Further investigation is necessary to clearly delineate the actual role of interparticle interactions in mixed suspensions. Figure 10 illustrates the combined effect of dosage and molecular weight of the PAA polymer along with that of Darvan C (a polyacrylate polymer of molecular weight 15 000). It would seem that Darvan C is effective only up
to about 20 ppm concentration; on the other hand, PAA polymers could assist in the formation of aggregates up to 100 ppm dosage. The aggregation index at a given concentration of polymer is uniformly higher in the case of higher molecular weight polymer. Moreover, the concentration at which the aggregation index flattens out is somewhat lower in the case of the higher molecular weight polymer. These trends show that an increase in the molecular weight (polymer chain length) leads to a higher degree of aggregation. Next, we examine two systems for which published data on electrokinetic sonic amplitude (ESA) in suspensions is available. Figure 11a from Pradip et al.15 shows the ESA measurements on alumina suspensions dosed with Darvan C surfactant, and Figure 11 b presents the aggregation index of these suspensions prepared under same conditions. The positions of the maximum aggregation index in each case agree well with the isoelectric points as read from the ESA curves. A similar comparison is presented in Figure 12 for the ESA data of Pradip et al.15 for zirconia (SY 5.2) suspensions with Betz 1190 as surfactant. Since it is a cationic polymer, the isoelectric point is shifted toward alkaline pH. The reasons we observe aggregation in the presence of a dispersant were discussed in our earlier publications.15,16 Essentially the isoelectric point of the system shifts to acidic or alkaline pH values depending upon whether anionic or cationic surfactant has been used. The shift in isoelectric point is a function of surfactant dosage. The observed maximum in the aggregation index thus also shifts with the dosage. What is a more interesting observation is that the aggregation index is higher at higher dosage of the polymer. While the shift in the pH due to electrosteric interactions is clearly understood,13-15,30
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more investigations are required to delineate the reasons why the aggregation index progressively increases with increasing dosage. Concluding Remarks Healy and co-workers31-33 and Pradip et al.15-17 have shown the importance of the isoelectric point in regulating the extent of dispersion both with and without the presence of surfactants. The yield stress of the suspension, which also characterizes the extent of dispersion, is found to be maximum at the isoelectric point. The yield stress in turn can be related to interparticle surface forces, the primary particle size distribution, the solid loading, and composition in the case of mixed suspensions.17,18,34,35 It turns out that a much simpler measurement of particle size distributions when converted to an aggregation index, as per the procedure outlined in the paper, also provides similar information, namely, the aggregation index is maximum at the isoelectric point. This single-value index associated with the suspension can be used to study the (30) Hackley, V. A. J. Am. Ceram. Soc. 1997, 80, 2315. (31) Leong, Y. K.; Katiforis, N.; Harding, D. B. O’C.; Healy, T. W.; Boger, D. V. J. Mater. Process. Manuf. Sci. 1993, 1, 445. (32) Leong, Y. K.; Boger, D. V.; Scales, P. J.; Healy, T. W.; Buscall, R. J. Chem. Soc., Chem. Commun. 1993, 7, 639. (33) Leong, Y. K.; Scales, P. J.; Healy, T. W.; Boger, D. V. J. Am. Ceram. Soc. 1995, 78, 2209. (34) Kapur, P. C.; Scales, P. J.; Boger, D. V.; Healy, T. W. AIChE J. 1997, 43, 1171. (35) Scales, P. J.; Johnson, S. B.; Healy, T. W.; Kapur, P. C. AIChE J. 1998, 44, 538.
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effect of various process parameters such as the kind of surfactant used, its dosage and molecular weight, etc. Particle size analysis in suspensions is a particularly attractive method for characterizing mixed suspensions. The size analysis technique for studying dispersion/ aggregation in suspensions serves as a powerful tool in providing guidelines about suspension conditions to be used while processing powder slurries for ceramic applications. As mentioned in the Introduction, a good dispersion of slurries is needed primarily to prevent regions of local agglomeration in the green ceramic compact. In general to obtain a high-density compact the slurry from which it is slipcast, must be well dispersed. It has been observed that when dispersants are used at high concentrations, the compact formed often has cracks and is always of a lower green density. It may be inferred from these observations that such undesirable effects could be due to the presence of aggregates in the slurry. Our results show that the prevailing degree of aggregation in suspensions may be conveniently expressed in terms of the aggregation index. Acknowledgment. Financial support for this work from US-India fund under the auspices of Department of Science and Technology (DST), India, and National Institute of Standards and Technology (NIST), USA, is gratefully acknowledged. Our sincere thanks to Professor E. C. Subbarao for his keen interest in the work and for critically reviewing this manuscript. LA980744E
