Ind. Eng. Chem. Fundam. 1984,2 3 , 406-410
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Particle Clusters in Concentrated Suspensions. 1 Experimental Observations of Particle Clusters Alan L. Graham‘ Universify of California, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
R. Byron Blrd Chemical Engineering Department and Rheology Research Center, Universiw of Wisconsin-Madison, Madison, Wisconsin 53706
This article is the first in a series that combines experimental observations and theory to study particle clusters and how they relate to transport properties in concentrated suspensions. The primary experimental observation is that particles in sheared suspensions, where only hydrodynamic forces exert an appreciable effect, form clusters that are continuously created and destroyed. These clusters translate and rotate as the suspension is sheared. By using a new homogeneous flow apparatus, we measured directly the frequency distributions of sizes of clusters of spheres in dilute suspensions and obtained partial information on concentrated suspensions from observations of clusters of labeled spheres in transparent suspensions. We then determined, over the range of our experimental data, the effects of volume concentration of particles and Reynolds number on the cluster-size distribution in sheared suspensions.
I. Introduction In homogeneous shear flow experiments, we observed that particles in both dilute and concentrated suspensions form clusters that are continuously created and destroyed. These clusters translate and rotate as the suspension is sheared. The experiments were conducted in a new homogeneous flow apparatus (HFA) whose design, unlike that of concentric cylinder devices, allowed the observation of large spheres in a region of the shear field where no curvilinear flow exists. We selected the large spheres to ensure that only hydrodynamic forces appreciably affected the motion and interaction of the particles. The suspensions consisted of these large spheres, which were uniform, solid, and neutrally buoyant in Newtonian liquids. Three television cameras, two of which were focused on a single point in the interior of the apparatus and one that was used to record the motor-driven belt speed, produced the video film. These films were used to define the motion of the particles in three spatial dimensions. The use of split-screen recording ensured the simultaneity of the two views of the interior of the apparatus. In the dilute suspensions, all of the particles were observable and the frequency distributions of sizes of clusters of spheres were measured directly from the video recordings. The more concentrated suspeiisions were rendered transparent by matching the refractive index of the suspending liquid to that of the particles. Partial information on the particle clusters in the concentrated suspensions was obtained from observations of the clusters of a small fraction of opaque particles added to the transparent suspensions. The effects of the concentration of spheres and the Reynolds number on the frequency distribution of sizes of clusters of opaque particles were determined. In later papers in this series, we shall see how the partial information obtained experimentally in the concentrated suspensions is combined with the Maximum Entropy Principle and Shannon’s Information Theory to infer the size distribution of the particle clusters of all of the spheres. This combination of experimental observations and sta0 196-43 13/84/1023-0406$0l.50/0
tistical inference will then form the basis for the prediction of the macroscopically observed viscosity of the suspension. This paper, part 1,describes the experimental program, the data collected, the data reduction procedures, and the statistical analysis of the data. 11. Experimental Apparatus and Procedures The experimental system, described in detail by Graham (1980),consisted of the HFA, the suspension components, and the video recording system. A drawing of the apparatus and camera positioning during the data recording is shown in Figure 1. A. The Homogeneous Flow Apparatus. The HFA consists of two flat, parallel belts and a containing vessel made from polymethyl methacrylate (PMMA) and aluminum bracing plates. The two fiber-reinforced neoprene belts are coupled with a chain drive so that their motion is parallel, but opposite in the center section of the HFA. The motor manufactured by the PMI Division of Kollmorgan Corporation produces smooth, steady motion of the belts. The entire apparatus was grounded. The containing vessel was designed so that the fluid is subjected to a constant shear rate everywhere except for curvature corrections at the ends. This geometry was first suggested by Prud’homme (1978). The homogeneity of the pure fluid velocity profiles in the HFA interior was established by measuring, as a function of position, the velocity of small air bubbles. Smooth, laminar flow was observed throughout the apparatus. A t a belt-width/gap-width ratio of 0.5, the velocity profile was linear in the center 65% of the belt width. In addition, the flow was fully developed at two gap widths from the ends of the belt loop (the gap-length/gap-width ratio was approximately 13). The remaining approximately 60% of the gap length was in homogeneous shear flow and was suitable for observations of particle motions and interactions. B. The Suspension Components. The fluid was a UCON oil/tetrabromoethane solution. The UCON oil 50-HR-5100, a product of Union Carbide Corporation, is 1984 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984
Figure 1. Homogeneous flow apparatus (HFA). Observations of particle motions and interactions in homogeneous shear flow were made at the midpoint of the external dimensions of the HFA. Two standard Sony color cameras, 45" apart in the same plane, provided information on all three components of the position of a sphere. A third camera monitored the belt speed by recording the passage of belt markings along a ruler affixed to the outside of the HFA. The flow field was fully developed 150 mm from either end of the HFA. In this region, the inhomogeneities produced by the containing walls extended 15 mm into the flow field, leaving the center 65% of the apparatus depth in HSF with linear velocity gradients (4=2%). The apparatus held 11.2 L of pure liquid, of which 7.65 L was available to the suspension.
a polyalkylene glycol that is stable, nonhydroscopic, and soluble in cold water. T h e 1,1,2,2-tetrabromoethane(TBE) was obtained from Aldrich Chemical Company. Like all halogenated hydrocarbons, TBE is a potential mutagenic agent, and therefore it was necessary to avoid skin contact or breathing the vapors for prolonged periods in poorly ventilated areas (Brem et al., 1974). TBE was added to the UCON oil until the refractive index of the solution was 1.491. The resultant 38 wt % TBE solution in UCON oil had a specific gravity of 1.387 at 22 "C and 1.372 at 31 "C. The UCON/TBE solution was Newtonian, with no detectable normal stresses on a Lodge stress meter. Viscosity values of 2.83 Pa s (28.3P) at 21.6 "C, 1.72 Pa s (17.2P) at 29.0 "C, and 1.11 Pa s (11.06P) at 38.0 "C were determined. The suspended particles were PMMA spheres with diameters of 6.4 mm (0.25in.). They have a specific gravity and refractive index of 1.19 and 1.491,respectively. The diameters of the spheres were uniform to within &0.5%. The small density difference between the PMMA and the UCON/TBE solution produced an experimentally measured rise velocity of 13 mm/s for the spheres. To help ensure that this value was negligible, the smallest belt velocity studied was more than an order of magnitude larger than the rise velocity of the spheres. In preparing the suspensions, the opaque spheres for making a 4 = 1% suspension, where 4 is the volume concentration of particles in suspension,were taken equally
407
from four different colors. Except for opacity, the opaque spheres had the same dimensions and physical properties as the transparent spheres. One in three opaque spheres was individually labeled with three press-on letters or symbols to allow easy identification of individual spheres from any angle. The spheres were then sealed with a thin coat of cyanoacrylate adhesive. After observations were completed on the #IJ = 1% suspensions, the number of opaque particles was held constant, and the overall concentration of spheres was raised by introducing transparent spheres until the desired concentration level was obtained. Because the refractive index of the UCON/TBE solution matched that of the transparent spheres, the additional spheres could not be seen. By matching the refractive index of most of the particles in concentrated suspensions, we ensured that the HFA interior was observable and that at least partial information could be obtained on particle interactions in concentrated suspensions. C. The Data Recording System. Particle motions were recorded on video tape as the suspension was sheared. As shown in Figure 1, three cameras were used to record an experiment. Two color cameras, 45" apart in the same plane and focused on the same point in the flow field, provided information on all three components of the relative position of a particle. The third camera, a black and white unit, was used to record the belt velocity. Two cameras are required to record particle motion in three dimensions. The origin of the coordinate system was the midpoint of the external dimensions of the HFA (Figure 1). The origin of the Cartesian coordinate system was the focal point of cameras 1 and 2. Camera 1 sighted along the z axis and camera 2 along the line defined by z = x in the plane y = 0. The x and y components of position were taken directly from camera 1observations. The z component of position was related by elementary applications of Snell's law to the vertical component of position as seen by camera 1,z, and the vertical component of position as seen by camera 2, x2, by z = (2.6253~~ 1.8564~)(see Appendix A for details). Camera 3 was aligned along the z axis to continuously monitor belt speed by recording the passage of a tape strip along a ruler taped to the sidewall of the HFA. All three cameras were started simultaneously and operated continuously. Observations from cameras 1 and 2 appeared simultaneously on the video tape, using the split-screen video. Electronic switching of camera 3 was used to record occasional checks of the belt's speed. The optics on the color cameras gave a very limited depth of field. When the cameras focused on the center of the flow field, the spheres located near the HFA walls were out of focus and, hence, could not be distinguished and were ignored in the data reduction. D. Data Collection and Reduction. The two experimental parameters controlled in this study were the volume fraction 4 of spheres and the sphere Reynolds number, Re (Re = a 2 ? p / p , where a is the radius of one of the spheres, 9 is the macroscopic shear rate, and p and p are the suspending liquid density and viscosity, respectively.) We refer to suspensions with a particular 4 and Re by experiment numbers as defined in Table I. Each of the three values of 9 studied for the 4 = 1% suspension (experiments 1-3) was recorded for 1 h. The belt speed and temperature were checked every 10 min. After these experiments, pure liquid was removed from the HFA through an exit port covered by a screen. Sufficient transparent particles to yield 4 = 10% were then added. The 9 for the 4 = 10% suspension was recorded on video tape for 1 h. This procedure was repeated for the 4 = 20%
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Table I. Experiment Descriptions and Summary of Experimental Results
+
vol % spheres: vol % opaque spheres; sohere Revnolds numbers.'+ Re x l@ probshilitf of actually observing an opaque sphere in a cluster of size i, p p i = l ~
~~
~~
~.
~
~
i=2
i=3 i=4 i=5 i>6 total no. of spheres observed no. of subsets of approximately 325 sphere observations
1 1.0 1.0 1.10 0.719 0.142 0.052 0.037 0.026 0.024 1956 6
2.72
eapt 110. 3 4 .. . 1.0 lu.u 1.0 1.0 4.21 2.33
0.704 0.172
0.752 0.161
0.066
0.060
0.023 0.014 0.021 2007 6
0.019 0.002
2
1.0 1.0
2135 6
1.0 2.53
6 20.0 1.0 6.74
0.870 0.107 0.023
0.860 0.115 0.023 0.002
0.849 0.128 0.019 0.004
2095
2164
7
7
2020 6
5 ...
2u.u
"The spheres were made from polymethyl methacrylate (n = 0.635 mm). The Newtonian suspending fluid was 38 w t % 1,1,2,2-tetrahromoethane, in UCON oil 50-HB-5100 from Union Carbide Corp. with a solution density of 1.19 g/cm3 and a solution viscosity of 28.3 P at 21.6 'C. bRe = (a'?p)/p, where a is the radius of the spheres, 4 is the macroscopic shear rate, and p and ~rare the suspending liquid density and viscosity, respectively. 'The observed probability, P,", in experiments 1-3 is equal to the actual probability of a cluster of size i existing, p i , because in the suspensions with = 1% all of the spheres are observable.
+
suspension at each of the two values of i. studied. When suspensions were placed in the HFA, we observed that the spheres translated and rotated in clusters that were continuously formed and destroyed, with lifetimes of 0.1 to 2.5 s. The objective of the data reduction was to determine the probability distribution of the cluster sizes of opaque spheres. The volume fraction of opaque spheres was equal to 1% in all suspensions. A cluster was defined as a collection of spheres that are observed to be in contact on both screens at the instant of observation. Only those particles that could be identified simultaneously on both sides of the split screen were counted. The particles that were not sharply in focus were not counted because they were in the inhomogeneous flow region near the walls of the HFA. When the video tape was stopped at a particular frame, 16 to 30 in-focus spheres could be observed simultaneously on both screens (Figure 2). The first and most difficult step in the data analysis consisted of identifying individual particles on both sides of the split screen. Using different colors and identifying individual spheres with press-on letters facilitated this critical step in the data reduction. After the particles bad been identified, the number of clusters and the number of spheres in each cluster were determined and recorded. The film was advanced 2.5 s, and the procedure was repeated. We note that even at the smallest shear rate, an almost complete rearrangement of the spheres in clusters occurred every 0.5 s. A subset of data from each experiment was determined from analyzing 16 frames of the video films. The results of these approximately 325 sphere observations were tabulated in terms of the number of opaque spheres in clusters of size i. In the suspensions in which 4 equals 1% all spheres were opaque. In these dilute suspensions, i varied from 1to 11. In the suspensions in which 4 equals 10 and 20%, only 1in 10 or 20 spheres was opaque, and i varied from 1to 4. Six or seven of these s u h t s of approximately 325 sphere observations were taken on each tape analyzed. The total of approximately 2100 opaque sphere observations for each experiment was taken from less than 5 min of video tape. During this period, the suspension temperature was constant to within f0.2 "C. Examination of the belt gear teeth with a strobe light revealed less than a 1% variation in the belt's speed. The results of the data reduction are presented in Figure 3. The volume concentration of opaque spheres, @,,paquo, was equal to 1% in all experiments. The f w e s presented are in terms of the probability, pp. of observing an opaque sphere in a cluster of size i. Note that they are not
CAMERA I
I
CAMERA 2
Figure 2. Split-screen video data recording. Tracing of split-screen video data recording. The view on the left-hand screen is from camera 1 and the view on the right-hand screen is from camera 2. Particles have the same relative horizontal position on both screens; this is essential in the particle identification step of the data reduction. Spheres are classified as a cluster only if they are ohsewed to be touching in hoth screens. Clusters in this frame are labeled with the same Greek letter on each screen.
presented as the frequencies gia of observing an opaque cluster of size i. The superscript "on on the probabilities indicates measured opaque sphere observations. The probabilities without the superscript refer to all of the spheres. In the dilute suspensions where all of the spheres are opaque, these two probabilities are identical. As will be discussed in part 2 of this series, in the concentration suspensions the pya provide the experimentally measured constraints for inferring the pi's. A summary of the raw experimental data obtained from the data reduction is available elsewhere (Graham, 1980). 111. Data Analysis and Discussion
The probability distributions of observing a sphere in a cluster with i members, pp, were analyzed and compared statistically. The results are summarized in Figure 3, and details of the statistical analysis are presented in AppendiK B. In the dilute suspensions (4 = 1%in experiments 1-3), we determined with 95% confidence that the distributions were drawn from different populations. Both the means 15p and the variances sj' of the distributions in experiment J were found to be inverse functions of Re. As Re increased
Ind. Eng. Chem. Fundam.. Vol. 23. No. 4, 1984 408 e-
+FRONT
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SURFACE OF HFA
vi SYSTEM 1
+ !
SYSTEM 2
SYSTEM 3
75
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f-'
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NUMBER OF SPHERES IN A CLUSTER, i
Figure 3. pmbability,pp, of observing an opaque sphere in a cluster of size i VB. the cluster size i. In all of the experiments, the volume concentration of opaque spheres is 0.01. In experiments 1-3, all of the spheres are opaque. In experiments 4 6 , the probability distributions are for the opaque fractions of the spheres. Details of the statistical calculations are found in Appendix B.
26670, p" and sj' decreased 16% and 71%, respectively, in the ddute suspensions. The decreases in p; and sj' were caused primarily by rearrangements between clusters with two or more members because the probability of the appearance of a sphere in a cluster of one, pI0,was not statistically different in experiments 1-3. The probability distributions may be reformulated in terms of the probability that a sphere will occupy a cluster with i - 1additional particles, ji.lo = pp. The mean of fp in experiment j , fjo, may be interpreted as the average number of neighbors a particle has in a clu_sterin experiment j . Box (1980) pointed out to us that f j o approaches the value of the variance of the f; distributions at high Re. The quantity 7; may be interpreted as the average number of neighbors in a cluster in experiment j . The ratios of f; to the variance of the distribution sjZ are 0.36.0.44, and 0.72 for experiments 1,2, and 3, respectively. If the mean equals the variance, the number variable is consistent with a Poisson distribution in which the physical process responsible for the distribution is random (Box, 1980). Over the range of data collected here, the trend is toward a random clustering of spheres as Re is increased. A t low values of Re, where viscous forces play a more significant role, the distributions could possibly be modeled by modified Poisson distributions, such as contaminated distributions, or by Neyman's contagious distributions. These distributions are discussed by Patil and Boswell (1974). As will be discussed in part 2 of this series of papers, a third alternative is to model these distributions
I
-z112=1 491
Figure 4. Determining the z component of position from the vertical components of the relative positions as seen by cameras 1 and 2. The incoming rays are considered parallel because the two cameras are approximately 1 m (3 ft) away and the distance from the front surface to the particles is approximately 50 mm (2 in.).
by using the Maximum Entropy Principle and Information Theory. In the more concentrated suspensions, 4 = 10% and 4 = 20%, only a small fraction (1 in 10 and 1 in 20, respectively) of the spheres were opaque to allow observations in the interior of the flow field. The partial informatiop from observations of the clusters of opaque spheres was sufficient to differentiate statistically between the frequency distributions p ; in the 4 = 10% and the suspensions with 4 = 20%. but not between those in the suspensions with 4 = 20% taken at different Re (experiments 5 and 6). Note that the means, p,, and the variances, ,:s of the p ; frequency distributions for the 4 = 20% suspensions show the same qualitative functionality on Re as the suspensions with 4 = 1%,although statistical s u p port is lacking. IV. Conclusions We observed that in sheared suspensions particles form in clusters that are continuously created and destroyed. These observations are made in Suspensions of neutrally buoyant spheres in Newtonian liquids designed in such a manner that only hydrodynamic forces are responsible for the particle clusters. Probability distributions of sizes of clusters of spheres were used to summarize the particle cluster data from a total of over 12000 sphere observations. Even in the dilute suspensions (4 = 1%) studied, over 25% of the spheres were observed in clusters with two or more members. Over our limited range of data there is a tendency for the average cluster size and the breadth of the distributions to decrease and for the distributions to become more random as the Reynolds number increases. In the next paper, we will discuss the inference of cluster size distributions for all spheres in the concentrated suspensions, using information theory and the partial information provided by observations of opaque sphere interactions. Later papers will deal with the linkage between microrheological observations and the macroscopic p r o p erties of suspensions. Acknowledgment The authors are indebted to G. E. Cort of the Lo8 Alamos National Laboratory for modeling the flow field in the apparatus. Professor Robert K. Prud'homme of Princeton University is acknowledged for several valuable discussions pertaining to the design of the apparatus. The staff of the Rheology Research Center, University of WisconsinMadison, including Professor Authur S. Lodge, Richard Williams, Ronald McCabe, MI. Dai Gance, Norman L. Johnson, and Barbara J. Yarusso, made contributions to the experimental design and procedures. We are grateful
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to Robert F. Perras and Richard Geier of the Engineering Audio Visual Service, University of Wisconsin-Madison, for their advice and assistance in video recording and for providing the editing facilities for the data reduction. We thank Dr. Raymond D. Steele of the Los Alamos National Laboratory and Professor George E. P. Box of the University of Wisconsin for their thoughtful consultations concerning the statistical aspects of this study. National Science Foundation Grant ENG-78-06789 and a grant from the Vilas Trust Fund of the University of Wisconsin provided financial assistance during the course of this work. Those portions of this work accomplished at the Los Alamos National Laboratory were sponsored by the U.S. Department of Energy Contract W-7405ENG-36 with the University of California.
Appendix A Relative Positions of Particles from Two Camera Views. As shown in Figure 1, the relative horizontal (y) and vertical ( x ) positions seen by camera 1 give directly two of the components of the particles' relative position. The third component of position, the z component, is determined by combining measurements of the particles' vertical separation is seen by camera 1,x , with the vertical separation as seen by camera 2, xq. One observes by Snell's law (see Figure 4) q1 sin d1 = qz sin 9, 64-11 where 7, = 1.000 = the refractive index of air, q2 = 1.491 = the refractive index of the suspending liquid and PMMA, and 8, = 45O. Solving for d,, we determine d2 = 28.31'. From geometric considerations, we see x , cos 92 = x , (-4-2) and xi
cos 91 = xp
Eliminating xi between the equations yields X Z = (COS B,/COS B ~ ) x , = 0.803174~~
(A-3) (A-4)
We also see that x , = z sin d2
+ x cos O2
Substituting eq A-4 into eq A-5 yields x 2 = (cos B,/cos d,)(z sin 9, + x cos 8,) = cos dl(z tan 0, + x )
(A-5)
$J
j
(A-6)
This equation is solved for z to give z = [(X~/COS e,) - X I cot e, and after substituting numerical values for d and B2, we find = 2.6253~2- 1.8564~
at the 95% confidence limit indicate that the distributions for experiments 1-3, the I#J = 1% suspensions, were taken from different populations (that is, the distributions were statistically different). The hypothesis that the distributions for experiments 5 and 6, the = 20% suspensions, were taken from the same population could not be rejected at a = 0.01 or at a = 0.05. The hypothesis that the frequency distribution of pi0 for experiment 4, the $J = 10% suspension, was the same as the distribution for experiment 6 was rejected at a = 0.05, but not at a = 0.01. Because many (approximately 2100) sphere observations were made on each experiment, a normal distribution (as opposed to a t-distribution) was used to test whether the mean of experiment i, p r , was statistically different from the mean of experiment j , p: (a pooled standard deviation was used for this test). At the 95% confidence limit, the results of this statistical test for experiments 1-3 show that the means are statistically different and that pl0> pzo> p30. In experiments 4-6, this same analysis showed at a = 0.05 that p40and pE0,and p50 and psoare not statistically different but that p40 < p50. An F-test was used to determine whether the variances of experiment i and j , si2 and sf, respectively, were statistically different. The results showed that the si's for experiments 1-3 were all statistically different and that s12> s; > s3, at a = 0.05. In the more concentrated suspensions, it was determined that only :s and $62 were not statistically different ( a = 0.05) and that s42 < sg2. The subsets of approximately 325 data points were used in a pooled variance t-test to investigate the differences between the frequency of observing a cluster with only one member, p t , in experiments 1-3 and also to predict with 95% confidence the appearance of an opaque sphere in a cluster of size i, p?, in experiments 4-6. The results of the t-test indicate that at a = 0.05 the pl0's for experiments 1-3 are not statistically different. Nomenclature a = radius of a sphere f i = probability of a sphere having i - 1 nearest neighbors fj = the average number of nearest neighbors in experiment J gio = probability of observing a cluster of size i pi0 = probability of observing a sphere in a cluster of size i p j = the average cluster size in experiment j s j = variance of the distribution of cluster sizes in experiment
(-4-7)
Appendix B Statistical Tests on the Reduced Data. Statistical tests were performed on the measured distributions. The conclusions are sensible to the level of confidence 1 - a. Here a is the probability of rejecting a hypothesis, which is actually true. The techniques are described in standard references; for example, Miller and Freund (1977). A x2 test was used for the hypothesis that the distributions were taken from the same population. The results
Re = a z + p / p = sphere Reynolds number Greek Letters 4 = macroscopic shear rate p = viscosity of the suspending liquid p = density of the suspending liquid 4 = volume concentration of particles in the suspension Literature Cited Box, G. E. P. University of Wisconsin, personal communication, 1980. Brem, H. A.: Stein, 8.; Rosenkranz, H. S. Cancer Res. 1974, 3 4 , 2576. Graham, A. L. Ph.D. Thesis, University of Wisconsin, Madison, W I , 1980. Miller, I.; Freund, J. E. "Probability and Statistics for Engineers"; PrenticeHail, Inc.: Englewood Cliffs, NJ, 1977. Patii, G. P.; Bosweii, M. T. "Statistical Distributions in Scientific Work", Uoi. 2; Patii, G. P.; Kotz, S.; Ord, J. K.; Ed.; D. Reidei Publishing Co.: Boston, MA, 1974: OD 83-94. Prud'homme. R. K. Princeton University, personal communication, 1978.
Received for review February 1, 1983 Accepted March 15, 1984
