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Sedimentation, Péclet number, and hydrodynamic screening Kiley Benes, 1 Penger Tong, 2 and Bruce J. Ackerson 1 1 Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA 2 Department

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Sedimentation, Péclet number, and hydrodynamic screening Kiley Benes, 1 Penger Tong, 2 and Bruce J. Ackerson 1 1 Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA 2 Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Received 13 January 2007; revised manuscript received 7 May 2007; published 8 November 2007 The sedimentation of hard spheres in a Newtonian solvent is studied as a function of Péclet number in the low-concentration limit. Two functional forms for the sedimentation velocity as a function of particle concentration are realized in the limit of high and low Péclet numbers. We argue that a more ordered phase occurs for large Péclet numbers. Measurements of settling in sheared suspensions support these contentions. Recent explanations of sedimentation in suspensions are examined in light of these results. DOI: /PhysRevE PACS number s : s, Hj, a, Pp I. INTRODUCTION The sedimentation of a collection of uniformly sized spheres in a Newtonian solvent represents one of the simplest nonequilibrium processes. Sedimentation has long been recognized as a medical diagnostic tool 1 and as an analytic tool to determine macromolecular dimensions 2. These techniques typically determine macromolecular dimensions using the Stokes velocity U 0 =2 ga 2 / 9 for sedimentation measurements in the limit of infinite dilution. Here a is the sphere radius, is the solvent viscosity, g is the acceleration of gravity, and is the particle-solvent density difference. Batchelor 3 determined the first-order concentration correction to the Stokes velocity. He assumed i a low particle Reynolds number or the neglect of inertia, ii twobody hydrodynamic interactions, iii a random particle distribution in space, and iv the system size infinite in the direction transverse to the settling. The result is given in terms of the particle pair correlation function g r and the particle volume fraction : U = U , x 1 g x dx + 15 g x 2 dx, 1 = x where x=r/a with r being the interparticle separation. For a dilute hard-sphere interaction, the integrations are readily performed to find U = U Direct particle interactions influence the form of the pair distribution function g r. A repulsive interaction between particles decreases the sedimentation velocity U, while an attractive interaction increases the sedimentation velocity compared to the hard-sphere result given in Eq. 2. Mazur and van Saarloos 4 generalized the Batchelor result to larger particle concentrations. Experimentally the concentration dependence of the sedimentation velocity for hard spheres is known 5,6. Recent measurements 7 of sedimentation in dilute suspensions of charged spheres produced a different functional form from that given in Eq. 2. The experimental results produce the following nonanalytic form: 2 U = U 0 1 k 1/3 +, where k 2.8. Thies-Weesie et al. explain the above nonanalytic form using the rescaled mean spherical approximation RMSA pair distribution function for a screened Coulomb interaction between the charged particles. While they utilize the relationship between diffusion and sedimentation to predict sedimentation velocities, the same results can also be obtained using the Batchelor formalism. Intuitively, the repulsive charge interaction between spheres is so strong that the spheres maintain as large a separation from one another as possible. Under these conditions, the position of the primary maximum in the pair distribution function scales as the mean particle separation, a 1/3. This unanticipated concentration dependence in the pair distribution function modifies the concentration dependence given in Eq. 2 to produce the form in Eq. 3. While Batchelor produced the result given in Eq. 2 for a dilute random mixture of hard spheres, the prediction for a random array of fixed spheres gives 8 11 U = U 0 1 k 1/2 +, 4 and for a fixed array of ordered spheres the result is identical to Eq. 3 with the value of k depending on the crystal structure 10, Thies-Weesie et al. 7 provided intuition for these results: the change in concentration dependence going from Eq. 2 to Eq. 4 and to Eq. 3 is due to the successive switching off of Brownian motion and the positioning of spheres on an ordered array. Each step deprives particles of possibilities to screen each other from the backflow and hence increases the friction per particle. Some time ago, Oliver 15 compiled data for the sedimentation velocity measured for dilute suspensions of spheres larger than those quoted above. Remarkably these suspensions produced the same concentration dependence as that given in Eq. 3. Oliver rationalized this concentration dependence by arguing that the interparticle separation is governed, not by the particle diameter, but again by the relationship a 1/3. The particles find themselves as far away from one another as possible. But Oliver gives no mechanism producing this order. Presumably these large particles are hard spheres with little direct interaction, as compared to the charged spheres discussed above. Surely these /2007/76 5 / The American Physical Society BENES, TONG, AND ACKERSON dilute suspensions are not crystalline arrays but are random distributions of freely moving hard spheres. Systems composed of larger or more dense hard spheres exhibit a variety of strange or seemingly contradictory behavior. Tory et al. 16 presented a variety of observations for monodisperse particle suspensions that demonstrate this remarkable complexity of behavior. The suspensionsupernatant interface becomes diffuse in dilute suspensions and falls with a rate that is remarkably constant and less than the Stokes velocity U 0 15, The interior of the suspension shows large particle velocity variation up to 4 times faster than the mean settling velocity 15,17, Large clusters of spheres settle more rapidly with fluid flowing around rather than through them The variance of particle velocities increases rapidly with volume fraction in the range from 10 3 to ,24 and as the container to particle size ratio increases 17. Similarly, the mean internal settling velocity increases with volume fraction and the container to particle size ratio. The mean internal velocity is greater than the Stokes velocity and hence is also greater than the suspension interface velocity 17,20,21,24. The volume fraction of the particles evidences a gradient at the suspension-supernatant interface. This gradient grows with time so that the interface becomes less distinct 19. In addition to the study of mean settling velocities, velocity fluctuations received renewed and intense attention after Calflisch and Luke 25 showed theoretically using the Batchelor assumptions that the velocity fluctuations depend on the linear dimension of the container and grow without limit as the sample container grows to infinite size. The velocity fluctuations have been studied experimentally and theoretically 20,33 35 and by computer simulation Experiments have not yet resolved whether the velocity fluctuations can reach a steady state independent of container size 27,29 or continue to evolve 31,41. Theoretically, it is recognized that horizontal boundaries at the top and bottom of a suspension and the presence of the suspension-supernatant boundary strongly influence the time evolution of velocity fluctuations. From computer simulations Nguyen and Ladd 42 argue that the suspension becomes more uniform as large-scale particle density fluctuations rise or sink out of the suspension. As the density fluctuations drain out, the velocity fluctuations decrease in magnitude. Alternatively, Mucha et al. 43 argue that velocity fluctuations decrease in magnitude as the suspension becomes stratified. The stratification, however slight, stabilizes the movement of density fluctuations and suppresses velocity fluctuations. In the experiment presented here, we measure the mean sedimentation velocity of monodisperse suspensions of hard spheres as a function of the Péclet number, which characterizes the strength of convection to that of diffusion. When the Péclet number is small, Brownian motion is significant and the particle configurations are continuously randomized, such that the Batchelor result is expected to hold. When the Péclet number is large, convection dominates, and for reasons described above the suspension becomes more uniform or stratified as compared to a random system. This should be evident in the mean settling velocity as well as in velocity fluctuations. Care is taken to eliminate any direct interaction TABLE I. Particle samples used in the present experiment and in literature data. Sample a m Pe Manufacturer and reference PS ± Bangs Laboratories PS ± Bangs Laboratories PS3 7.45± Bangs Laboratories PS4 2.55± Bangs Laboratories PS5 1.05± Bangs Laboratories CP1 0.38± Ref. 50 PMMA Ref. 15 between particles to avoid the effect of the screened Coulomb potential for charged particles, as measured by Thies- Weesie et al. 7. To further verify the thesis that particle ordering produces the nonanalytic concentration dependence, new sedimentation measurements are made for suspensions undergoing a simple shear flow, which is applied to destroy local and global interparticle structures. II. EXPERIMENT A. Measurement of the mean settling velocity Five polystyrene particle samples of different sizes are used in the experiment. They were purchased from Bangs Laboratories Inc. Dynaspheres and are suspended in water. In Table I we give the values of the radius a and the Péclet number Pe of each particle sample. The values of a are provided by the manufacturer. The polydispersity ratio of the standard deviation to the mean size of each sample is 2% or less. The Péclet number is given by Pe=aU 0 /D 0 = 4 /3 ga 4 / k B T, where D 0 =k B T/ 6 a is the Stokes- Einstein diffusion constant for individual particles. The Péclet number measures the importance of convection compared to diffusion. Because the particles as received are charge stabilized against aggregation by added surfactant, aliquots of different volume fraction particles are prepared using Barnsted de-ionized water 10 M with added sodium chloride NaCl to screen the interparticle charge interactions and sodium dodecylsulphate SDS, 0.05% by weight to maintain charge stabilization, albeit at very small screening lengths. The suspension settling velocity was measured, and a known amount of salt was added. The settling velocity was remeasured, correcting for changes in the solvent density. The process was repeated until the measured settling rates saturated. For salt concentrations 0.1 M and greater, the measured sedimentation velocities for a given particle size saturated to the values reported here. We conclude that under this condition the ionic screening is sufficiently great that the only direct interaction between the particles is well represented by the hard-sphere interaction potential. The samples fill 1.5-ml or 5-ml screw-top vials, where the sample heights are 3.5 cm or 4.5 cm, respectively. Even with these small volumes the ratio of the smallest cell dimension to the particle radius is greater than 350. The effect of con SEDIMENTATION, PECLET NUMBER, AND tainer boundaries on the settling velocity of a sphere at the center of a spherical volume, having a radius 350 times the sphere radius, is less than 1% different from the velocity for an infinite volume 18. In Bruneau et al. 44, Eq. A5 provides a correction for wall effects by integrating over the container dimension. The magnitude of this correction is less than 2% for the largest particles we used and much less than that for the rest. The samples are mixed in the sealed vials by slow tumbling end over end using a mechanical mixer. They are also tumbled by hand prior to measurement. After mixing, these vials are placed in a well stirred water bath to control temperature to 20±0.1 C, but more importantly to eliminate temperature gradients 45. Each particle size series has samples with volume fraction in the range from to The sedimentation velocity of each sample is determined by measuring the settling speed of the particlesupernatant interface, which is measured to within 10 m using a cathetometer. Typically five measurements are taken per sample over a period up to a few hours. All height versus time measurements show linear dependence R with typical precision of 5% for the fitted slope from the repeated measurements. We determine the density of polystyrene particles by measuring the sedimentation velocity of dilute samples in known mixtures of D 2 O and H 2 O, the former having a density greater than the particles and the latter a density less than the particles. For a mixture of solvents corresponding to density match with the particles, the sedimentation velocity is zero. From the sedimentation measurements in mixtures of different D 2 O mass fraction, we find that the density match occurs at the D 2 O mass fraction of and corresponds to a particle density p =1.052 g/cm 3, assuming volume conservation on mixing the two forms of water. This value of p agrees well with the literature value for polystyrene latex spheres. Pyncometer measurements give the densities of various salt, soap, and water mixtures used in the experiment. These values are used in the calculation of the Péclet numbers shown in Table I. B. Measurement of the mean settling velocity under a shear Polystyrene particles having a radius a=10.15 m PS2 in Table I are suspended at the volume fraction = in water with added NaCl and SDS as described above. The suspension is contained in a Couette shear cell 46. The shear cell is composed of two concentric cylinders. The outer radius of the inner cylinder is R 1 =1.45 cm, and the gap between the cylinders is h=0.58 cm. The height of the sample is 4.5 cm. The inner cylinder is powered by a stepping motor having 200 steps per revolution with the pulse rate under software control. The rotation is smooth for large angular frequencies, but is a pulsed motion at low rates of rotation. Therefore, data at selected low rotation rates are reproduced with a continuously driven dc motor. The shear rate varies across the gap, and we give the rate at the inner cylinder wall. Because the inner wall is moving, the suspension is subject to the Taylor-Couette instability 47. The critical rotation rate for the onset of the instability is c =41.3 / h 3/2 R 2 1/2, where h=r 2 R 1 and is the kinematic viscosity of the fluid. For our cell with inner radius R 1 =1.45 cm and outer radius R 2 =2.03 cm, we have c =0.67 rad/s. We do in fact observe the onset of this instability at the predicted value of c. The interface between the settling suspension and clear supernatant above tips either towards the inner or outer cylinder as it passes through different portions of the convective rolls. But for sufficiently small rotation rates, the interface remains flat, albeit a sensitive indicator of convective motion. The inner cylinder rests on a liquid mercury seal to exclude the suspension from the region beneath the inner cylinder. The mercury seal allows free rotation of the inner cylinder. Without the mercury seal, the suspension becomes unstable while settling. As the particles settle away from the bottom of the inner cylinder, a solvent bubble develops in contact with the cylinder end. The bubble eventually creeps up the vertical cylinder walls and makes the suspension unstable. The whole shear cell is immersed in a well stirred water bath to eliminate convection due to temperature gradients. The temperature is controlled to within 0.1 C for the duration of the measurements. Samples in the shear cell are mixed by raising the inner cylinder out from the sample and lowering back into the sample several times prior to measurement. Convection and turbulence mix the sample. The suspension height is monitored as a function of time to within 10 m using a cathetometer. The interface between the sedimenting suspension and supernatant becomes more diffuse with increasing hydrodynamic diffusion and polydispersity. The interface also becomes more diffuse in time with hydrodynamic diffusion being more dominant at early times 19. The reduced hydrodynamic diffusion constant Dˆ =D h / U 0 a depends on the volume fraction and has a value typically greater than unity and less than 15. For volume fractions less than 0.01 it is essentially unity 19. As a result the diffusion constant in the dimensionless diffusion equation becomes Dˆ a/h 0 or a/h 0 at small volume fractions. The interface spreading increases with increasing particle radius at fixed sample height H 0. In determining the settling velocity of the samples, we made direct visual observations of the interface rather than light transmission measurements 19,48. However, we restricted our measurements to the initial settling process in the upper quarter of the sample containers. Observation times were adjusted to account for the different settling rates. In this region the interface is the least diffuse and our measurements produce linear plots of interface height versus time to a high degree of precision. Numerically integrating the dimensionless drift-diffusion equation, using unity for the reduced hydrodynamic diffusion constant, indicates insignificant interface spreading less than a few percent of the height for all but the largest Pe value sample measured. III. EXPERIMENTAL RESULTS Figure 1 presents the measured sedimentation velocity U normalized by the Stokes velocity U 0. The solvent density is corrected for the amount of added salt using standard tables BENES, TONG, AND ACKERSON U/U U/U FIG. 1. Normalized sedimentation velocity U/U 0 as a function of volume fraction for hard spheres with the Péclet number Pe = solid squares, solid circles, stars, open triangles, open diamonds, open squares, and open circles. The solid line shows the function U/U 0 = , and the dashed line shows the function U/U 0 = / In addition, literature data are included for the lowest 50 and highest 15 Péclet numbers. The lowest Pe system Pe= is a suspension of m-radius hard spheres made of a copolymer core of methylmethacrylate and trifluoroethylacrylate stabilized by a poly-12- hydroxystearic acid coating and suspended in cis-decaline. The largest Pe sample Pe= is comprised of Kallodoc polymethylmethacrylate spheres with a mean radius a=80.5 mm in mixtures of glycerol and water. The large Pe particles settle more slowly compared to the smaller Pe particles. In the data presented here, the salt concentration is order 0.1 M, giving a Debye screening length less than 10 nm. The polystyrene particles, though charged to prevent coagulation, are highly screened and should behave effectively as hard spheres. Evidently the data shown in Fig. 1 separate into two different curves. For data with Pe 1, the approximation U/U 0 = solid line represents the data well and reduces to the Batchelor prediction U/U at small volume fractions. We notice that the approximation U/U 0 = fits the high- portion of the data slightly better and was also used in the literature 51. On the other hand, for Pe 1 the form U/U 0 = /3 gives a better fit to the data at small values of. To test the possibility that ordering influences the settling rate for large Pe suspensions, we perform sedimentation measurements on selected suspensions subjected to a shear flow. A shear directed perpendicular to the direction of settling will advect the particles along streamlines perpendicular to the direction of settling, moving some particles closer together along the direction of compression and pulling other particles further apart along the direction of extension. In general, a horizontal shear flow should only influence the interparticle ordering, but not mix the suspension vertically; any stratification in the suspension should remain. Figure 2 shows the measured sedimentation velocity U/U 0 as a func

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